Riemann surfaces of second kind and effective finiteness theorems
RRIEMANN SURFACES OF SECOND KIND AND EFFECTIVEFINITENESS THEOREMS
BURGLIND J ¨ORICKE
Abstract.
The Geometric Shafarevich Conjecture and the Theorem of de Franchis statethe finiteness of the number of certain holomorphic objects on closed or punctured Riemannsurfaces. The analog for Riemann surfaces of second kind is an estimate of the numberof irreducible holomorphic objects up to homotopy (or isotopy, respectively). This analogaddresses the problem of the validity of Gromov’s Oka principle.For any finite open Riemann surface X (maybe, of second kind) we give an effective upperbound for the number of irreducible holomorphic mappings up to homomotopy from X to thetwice punctured complex plane, and an effective upper bound for the number of irreducibleholomorphic torus bundles up to isotopy on such Riemann surface.If X σ is the σ -neighbourhood of a skeleton of a torus with a hole, then the number ofirreducible holomorphic mappings up to homomotopy from X σ to the twice punctured com-plex plane grows exponentially in σ . This can be interpreted as a quantitative statement ofGromov’s Oka principle. February 4, 20211.
Introduction and statements of results
The following statement, known as Geometric Shafarevich conjecture, has been conjecturedby Shafarevich [27] in the case of compact base and fibers of type ( g , Theorem A (Geometric Shafarevich Conjecture.)
For a given compact or puncturedRiemann surface X and given non-negative numbers g and m such that g − m > thereare only finitely many locally holomorphically non-trivial holomorphic fiber bundles over X with fiber of type ( g , m ) . A connected closed Riemann surface (or a smooth connected closed surface) is called oftype ( g , m ), if it has genus g and is equipped with m distinguished points. Recall that a closedRiemann surface with a finite number of points removed is called a punctured Riemann surface.The removed points are called punctures. Sometimes it is convenient to associate a puncturedRiemann surface to a Riemann surface of type ( g , m ) by removing the distinguished points. Aconnected Riemann surface is of first kind, if it is a closed or a punctured Riemann surface,otherwise it is of second kind. A Riemann surface is called finite if its fundamental groupis finitely generated, and open if no connected component is compact. Each finite connectedopen Riemann surface X is conformally equivalent to a domain (denoted again by X ) on aclosed Riemann surface X c such that each connected component of the complement X c \ X iseither a point or a closed geometric disc [14]. A geometric disc is a topological disc that liftsto a round disc in the universal covering of the closed surface X c with respect to the standardmetric on the universal covering. The closed Riemann surface X c and the embedding X ⊂ X c are unique up to conformal isomorphism. The connected components of the complement willbe called holes. Mathematics Subject Classification.
Primary 32G13; Secondary 20F36, 32H35, 32Q56, 57Mxx.
Key words and phrases. finiteness theorems, Riemann surfaces of second kind, 3-braids, torus bundles, Okaprinciple. a r X i v : . [ m a t h . C V ] F e b BURGLIND J ¨ORICKE
Theorem A was proved by Parshin [26] in the case of compact base and fibers of type( g , , g ≥ , and by Arakelov [2] for punctured Riemann surfaces as base and fibers of type( g , Theorem B (de Franchis).
For closed connected Riemann surfaces X and Y with Y ofgenus at least there are at most finitely many non-constant holomorphic mappings from X to Y . There is a more comprehensive Theorem in this spirit.
Theorem C (de Franchis-Severi).
For a closed connected Riemann surface X there are(up to isomorphism) only finitely many non-constant holomorphic mappings f : X → Y where Y ranges over all closed Riemann surfaces of genus at least . We may associate to any holomorphic mapping f : X → Y of Theorem B the bundle over X with fiber over x ∈ X equal to Y with distinguished point { f ( x ) } . Thus, the fibers areof type ( g , g , X to Y , but there are only finitely manydifferent holomorphic self-isomorphisms. Hence, Theorem B is a consequence of Theorem A,and Theorem B has analogs for the source X and the target Y being punctured Riemannsurfaces.The Theorems A and B do not hold literally if the base X is of second kind. If the base is aRiemann surface of second kind the problem to be considered is the finiteness of the numberof irreducible isotopy classes containing holomorphic objects. In case the base is a puncturedRiemann surface this is equivalent to the finiteness of the number of holomorphic objects. Formore detail see sections 2 and 3.We will prove finiteness theorems with effective estimates for Riemann surfaces of secondkind. The estimates depend on a conformal invariant of the base manifold. To define theinvariant we recall Ahlfors’ definition of extremal length (see [1]). For an annulus A = { ≤ r < | z | < R ≤ ∞} (and for any open set that is conformally equivalent to A ) the extremallength equals π log Rr . For an open rectangle R = { z = x + iy : 0 < x < b , < y < a } inthe plane with sides parallel to the axes, and with horizontal side length b and vertical sidelength a the extremal length equals λ ( R ) = ab . For a conformal mapping ω : R → U of therectangle R onto a domain U ⊂ C the image U is called a curvilinear rectangle, if ω extendsto a continuous mapping on the closure ¯ R , and the restriction to each (closed) side of R is ahomeomorphism onto its image. The images of the vertical (horizontal, respectively) sides of R are called the vertical (horizontal, respectively) curvilinear sides of the curvilinear rectangle ω ( R ). The extremal length of the curvilinear rectangle U equals the extremal length of R .(See [1]).Let X be a connected open Riemann surface of genus g ≥ m + 1 holes, m ≥ q . The fundamental group π ( X, q ) of X is a free group in2 g + m generators. We choose a system of generators as follows. The Riemann surface X isconformally equivalent to an open subset of a closed Riemann surface X c of genus g ([14]).Choose 2 g generators e j, , j = 1 , . . . , g, of the fundamental group π ( X, q ) whose imagesunder the homomorphism induced by inclusion X → X c are generators of the fundamentalgroup of X c . We may assume that each pair e j − , , e j, , j = 1 , . . . g, corresponds to a handle. IEMANN SURFACES OF SECOND KIND AND EFFECTIVE FINITENESS THEOREMS 3
Represent each pair of generators e j − , , e j, corresponding to a handle by simple closed loops α j , β j with base point q .Denote the connected components of X c \ X by C , C , . . . , C m +1 . Each component is eithera point or a closed disc. For each component C (cid:96) , (cid:96) = 1 , . . . , m, we choose the generator e g + (cid:96), of the fundamental group of X which is represented by a simple loop γ g + (cid:96) with base point q that is contractible in X ∪ C (cid:96) and divides X into two connected components, one of themcontaining C (cid:96) . We will say that γ g + (cid:96) surrounds C (cid:96) .We may choose the generators and the representatives so that the only intersection pointof any pair of loops is q , and when labeling the rays of the loops emerging from the basepoint q by α − j , β − j γ − j , and the incoming rays by α + j , β + j γ + j , then moving in counterclockwisedirection along a small circle around q we meet the rays in the order . . . , α − j , β − j , α + j , β + j , . . . , γ − k , γ + k , . . . . We denote the described system of generators by E , and fix it throughout the paper. Figure 1
Let ˜ X be the universal covering of X . For each element e ∈ π ( X, q ) we consider thesubgroup (cid:104) e (cid:105) of π ( X, q ) generated by e . Let σ ( e ) be the covering transformation corre-sponding to e , and (cid:104) σ ( e ) (cid:105) the group generated by σ ( e ). Denote by E j , j = 2 , . . . , , the setof primitive elements of π ( X, q ) which can be written as product of at most j factors witheach factor being either an element of E or an element of E − , the set of inverses of elementsof E . Denote by λ j = λ j ( X ) the maximum over e ∈ E j of the extremal length of the annulus˜ X (cid:30) (cid:104) σ ( e ) (cid:105) . The quantity λ ( X ) (for mappings to the twice punctured complex plane), or λ ( X ) (for (1 , E be a finite subset of the Riemann sphere P which contains at least three points.Let X be a finite Riemann surface with non-trivial fundamental group. A continuous map f : X → P \ E is reducible if it is homotopic (as a mapping to P \ E ) to a mapping whoseimage is contained in D \ E for an open topological disc D ⊂ P with E \ D containing at leasttwo points of E . Otherwise the mapping is called irreducible.In the following theorem we take E = {− , , ∞} . We will often refer to P \ {− , , ∞} as to the twice punctured complex plane C \ {− , } . Note that a continuous mapping froma Riemann surface to the twice punctured complex plane is reducible, iff it is homotopic to amapping with image in a once punctured disc contained in P \ E . (The puncture may be equalto ∞ .) There are countably many non-homotopic reducible holomorphic mappings from anyfinite open Riemann surface of second kind with non-trivial fundamental group to the twice BURGLIND J ¨ORICKE punctured complex plane (see the proof of Theorem 7 in [21]). On the other hand the followingtheorem holds.
Theorem 1.
For each open connected Riemann surface X of genus g with m + 1 holes thereare up to homotopy at most e πλ ( X ) ) g + m irreducible holomorphic mappings from X into Y def = P \ {− , , ∞} . Notice that the Riemann surface X is allowed to be of second kind. If X is a torus with ahole, λ ( X ) may be replaced by λ ( X ).A holomorphic (1 , ,
1) fiberbundle. The following lemma holds.
Lemma D.
A smooth (0 , -bundle admits a smooth section. A holomorphic torus bundle is(smoothly) isotopic to a holomorphic torus bundle that admits a holomorphic section. For a proof see [21].
Theorem 2.
Let X be an open connected Riemann surface of genus g with m + 1 holes. Upto isotopy there are no more than (cid:0) · · · exp(36 πλ ( X )) (cid:1) g + m irreducible holomorphic (1 , -bundles over X . We wish to point out that reducible ( g , m )-bundles over finite open Riemann surfaces canbe decomposed into irreducible bundle components, and each reducible bundle is determinedby its bundle components up to commuting Dehn twists in the fiber over the base point.Notice that Caporaso proved the existence of a uniform bound of the number of objects inTheorem A in case X is a closed Riemann surface of genus g with m punctures, and the fibersare closed Riemann surfaces of genus g ≥
2. The bound depends only on the numbers g , g and m . Heier gave effective uniform estimates, but the constants are huge and depend in acomplicated way on the parameters.Theorems 1 and 2 imply effective estimates for the number of locally holomorphically non-trivial holomorphic (1 , Corollary 1.
There are no more than e πλ ( X ) ) g + m non-constant holomorphic mappingsfrom a Riemann surface X of type ( g, m + 1) to P \ {− , , ∞} . Corollary 2.
There are no more than (cid:0) · · · exp(36 πλ ( X )) (cid:1) g + m locally holomorphicallynon-trivial holomorphic (1 , -bundles over a Riemann surface X of type ( g, m + 1) . The following examples demonstrate the different nature of the problem in the two cases,the case when the base is a punctured Riemann surface, and when it is a Riemann surface ofsecond kind. The first example considers holomorphic mappings from a torus with a hole tothe twice punctured complex plane. The second example considers mappings from a planardomain to the twice punctured plane. It shows that for Riemann surfaces X of second kindwe can not expect an upper bound for the number of irreducible holomorphic objects up tohomotopy depending only on the topological type of the base manifold X and of the fibers. Example 1.
There are no non-constant holomorphic mappings from a torus with one punctureto the twice punctured complex plane. Indeed, by Picard’s Theorem each such mapping extendsto a meromorphic mapping from the closed torus to the Riemann sphere. This implies thatthe preimage of the set {− , , ∞} under the extended mapping must contain at least threepoints, which is impossible.The situation changes if X is a torus with a large enough hole. Let α ≥ σ ∈ (0 , T α,σ that is obtained from C(cid:30) ( Z + iα Z ) , (with α ≥ α − σ and IEMANN SURFACES OF SECOND KIND AND EFFECTIVE FINITENESS THEOREMS 5 horizontal side length 1 − σ (i.e. we remove a closed subset that lifts to such a closed rectanglein C ). A fundamental domain for this Riemann surface is ”the golden cross on the Swedishflag” turned by π with width of the laths being σ and length of the laths being 1 and α . Proposition 1.
Up to homotopy there are at most e · π α +1 σ irreducible holomorphic map-pings from T α,σ to the twice punctured complex plane.On the other hand, there are positive constants c , C , and σ such that for any positivenumber σ < σ there are at least ce C ασ non-homotopic holomorphic mappings from T α,σ to thetwice punctured complex plane. Example 2.
There are only finitely many holomorphic maps from a thrice punctured Riemannsphere to another thrice punctured Riemann sphere. Indeed, after normalizing both, the sourceand the target space, by a M¨obius transformation we may assume that both are equal to C \ {− , } . Each holomorphic map from C \ {− , } to itself extends to a meromorphic mapfrom the Riemann sphere to itself, which maps the set {− , , ∞} to itself and maps no otherpoint to this set. By the Riemann-Hurwitz formula the meromorphic map takes each valueexactly once. Indeed, suppose it takes each value l times for a natural number l . Then eachpoint in {− , , ∞} has ramification index l . Apply the Riemann Hurwitz formula for thebranched covering Y = P → X = P of multiplicity lχ ( Y ) = l · χ ( X ) − (cid:88) x ∈ X ( e x − . Here e x is the ramification index at the point x . For the Euler characteristic we have χ ( P ) = 2,and (cid:80) x ∈ X ( e x − ≥ (cid:80) x = − , , ∞ ( e x −
1) = 3 ( l − ≤ l − l −
1) which ispossible only if l = 1. We saw that each non-constant holomorphic mapping from C \ {− , } to itself extends to a conformal mapping from the Riemann sphere to the Riemann spherethat maps the set {− , , ∞} to itself. There are only finitely may such maps, each a M¨obiustransformation commuting the three points.A domain in the plane is called doubly connected, if its fundamental group is isomorphic tothe free group in two generators. We consider now doubly connected planar domains containedin C \ {− , } . We take domains of a special form. Consider a skeleton S of C \ {− , } whichis the union of two simple closed curves γ and γ with base point zero, that surround − γ is contained in the cone { ρe θ : ρ ≥ , | θ | ≤ κ < π } and γ is contained inthe cone { ρe θ : ρ ≥ , | θ − π | ≤ κ < π } . Let σ be a small positive number and let Ω σ be the σ -neighbourhood of S . Proposition 2.
There are positive constants σ , C and C , depending only on γ and γ such that for each positive σ < σ there are up to homotopy no more than C e C σ irreducibleholomorphic mappings from Ω σ to the twice punctured complex plane.Vice versa, there are positive constants C (cid:48) and C (cid:48) such that for each σ < σ there are atleast C (cid:48) e C (cid:48) σ non-homotopic irreducible holomorphic mappings from Ω σ to the twice puncturedcomplex plane. The present results may be understood as quantitative statements with regard to Gromov’sOka principle. A mapping from a Stein manifold to a complex manifold Y is said to satisfyGromov’s Oka principle if it is homotopic to a holomorphic map. The target spaces Y , forwhich each continuous mapping from a Stein manifold to Y satisfies the Oka principle, arecalled Oka manifolds and received a lot of attention ([12], [9], [10]).The twice punctured complex plane is not an Oka manifold. Theorem 1 and Propositions 1and 2 give an estimate of the number of homotopy classes of mappings from particular Steinmanifolds to the twice punctured complex plane, that satisfy Gromov’s Oka principle. Theestimate depends on the conformal structure of the Stein manifold. BURGLIND J ¨ORICKE
The author is grateful to B.Farb who suggested to use the concept of conformal moduleand extremal length for a proof of finiteness theorems, and to B.Berndtsson for proposing thekernel for solving the ¯ ∂ -problem that arises in the proof of Proposition 1. The work on thepaper was started while the author was visiting the Max-Planck-Institute and was finishedduring a stay at IHES. The author would like to thank these institutions for the support. Theauthor is also indebted to Fanny Dufour for drawing the figures.2. Holomorphic mappings into the twice punctured plane
In this section we will prove Theorem 1. We first need some preparation.
The change of the base point.
Let X be a connected smooth manifold and let α be anarc in X with initial point x and terminating point x . Change the base point x ∈ X alonga curve α to the point x ∈ X . This leads to an isomorphism Is α : π ( X , x ) → π ( X , x )of fundamental groups induced by the correspondence γ → α − γα for any loop γ with basepoint x and the arc α with initial point x and terminating point x . We will denote thecorrespondence γ → α − γα between curves also by Is α .We call two homomorphisms h j : G → G , j = 1 , , from a group G to a group G conjugate if there is an element g (cid:48) ∈ G such that for each g ∈ G the equality h ( g ) = g (cid:48)− h ( g ) g (cid:48) holds. For two arcs α and α with initial point x and terminating point x wehave α − γα = ( α − α ) − α − γα ( α − α ). Hence, the two isomorphisms Is α and Is α differby conjugation with the element of π ( X , x ) represented by α − α .Free homotopic curves are related by homotopy with fixed base point and an application ofa homomorphism Is α that is defined up to conjugation. Hence, free homotopy classes of curvescan be identified with conjugacy classes of elements of the fundamental group π ( X , x ) of X .For two smooth manifolds X and Y with base points x ∈ X and y ∈ Y and a continuousmapping F : X → Y with F ( x ) = y we denote by F ∗ : π ( X , x ) → π ( Y , y ) the inducedmap on fundamental groups. For each element e ∈ π ( X , x ) the image F ∗ ( e ) is calledthe monodromy along e , and the homomorphism F ∗ is called the monodromy homomorphismcorresponding to F . The homomorphism F ∗ determines the homotopy class of F with fixed basepoint in the source and fixed value at the base point. Consider a free homotopy F t , t ∈ (0 , X to Y such that the value F t ( x ) at the base point x of the sourcespace varies along a loop. The homomorphisms ( F ) ∗ and ( F ) ∗ are related by conjugationwith the element of the fundamental group of Y represented by the loop. Moreover, since thefundamental group π ( Y , y ) with base point y is related to the fundamental group π ( Y , y )with base point y by an isomorphism determined up to conjugation we obtain the followingtheorem (see [13],[28]). Theorem E.
The free homotopy classes of continuous mappings from X to Y are in one-to-one correspondence to the set of conjugacy classes of homomorphisms between the fundamentalgroups of X and Y . Extremal length.
The fundamental group π def = π ( C \ {− , } ,
0) is canonically isomorphicto the fundamental group π ( C \ {− , } , q (cid:48) ) for an arbitrary point q (cid:48) ∈ ( − , α defining the isomorphism we take the unique arc contained in ( − ,
1) that joins q (cid:48) and 0. Thefundamental group π ( C \ {− , } ,
0) is a free group in two generators. We choose standardgenerators a and a , where a is represented by a simple closed curve with base point 0 whichsurrounds − a is represented by a simple closed curve with base point0 which surrounds 1 counterclockwise. For q (cid:48) ∈ ( − ,
1) we also denote by a j the generator of π ( C \{− , } , q (cid:48) ) which is obtained from the respective standard generator of π ( C \{− , } , − , a is the generator of π ( C \{− , } , q (cid:48) ) which is represented by a loop with base point q (cid:48) that surrounds − a is the generator of π ( C \ {− , } , q (cid:48) ) which is IEMANN SURFACES OF SECOND KIND AND EFFECTIVE FINITENESS THEOREMS 7 represented by a loop with base point q (cid:48) that surrounds 1 counterclockwise. We refer to a and a as to the standard generators of π ( C \ {− , } , q (cid:48) ).Further, the group π ( C \ {− , } ,
0) is canonically isomorphic to the relative fundamentalgroup π tr ( C \ {− , } ) def = π ( C \ {− , } , ( − , C \ {− , } with end points on the interval ( − , π tr ( C \ {− , } ) as fundamental group with totally real horizontal boundary values ( tr -boundary values for short). For an element w ∈ π ( C \ {− , } , q (cid:48) ) with base point q (cid:48) ∈ ( − , w tr the element of the relative fundamental group π tr ( C \ {− , } ) with totallyreal boundary values, corresponding to w . For more details see [18].Each element of a free group can be written uniquely as a reduced word in the generators.(A word is reduced if neighbouring terms are powers of different generators.) The degree (orword length) d ( w ) of a reduced word w in the generators of a free group is the sum of theabsolute values of the powers of generators in the reduced word. If the word is the identity itsdegree is defined to be zero. We will identify elements of a free group with reduced words ingenerators of the group.For a rectangle R let f : R → C \ {− , } be a mapping which admits a continuous extensionto the closure ¯ R (denoted again by f ) which maps the (open) horizontal sides to ( − , f represents an element w tr ∈ π tr ( C \ {− , } ) if for each maximalvertical line segment contained in R (i.e. R intersected with a vertical line in C ) the restrictionof f to the closure of the line segment represents w tr .The extremal length Λ ( w tr ) of an element w tr in the relative fundamental group π tr ( C \ {− , } ) is defined as Λ ( w tr ) def = inf { λ ( R ) : R a rectangle which admits a holomorphic map to C \ {− , } that represents w tr } . (1)For an element w ∈ π ( C \ {− , } , q (cid:48) ) and the associated element w tr we will also write Λ tr ( w )instead of Λ ( w tr ). In [18] (see also [19]) a natural syllable decomposition of any reducedword w in π ( C \ {− , } , q (cid:48) ) is given. Let d k be the degree of the k -th syllable from the left.(We consider each syllable as a reduced word in the elements of the fundamental group.) Put L − ( w ) def = (cid:88) log(3 d k ) , L + ( w ) def = (cid:88) log(4 d k ) . (2)Notice that L ± ( w − ) = L ± ( w ). We define L − (Id) = L + (Id) = 0 for the identity Id. We needthe following theorem which is proved in [18] (see Theorem 1 (cid:48) there). Theorem F. If w is not equal to a (trivial or non-trivial) power of a or of a then π L − ( w ) ≤ Λ ( w tr ) ≤ L + ( w ) (3) where the sum runs over the degrees of all syllables of w tr . Regular zero sets.
We need some facts concerning regular zero sets of smooth real valuedfunctions. We will call a subset of a smooth manifold X a simple relatively closed curve if itis the connected component of a regular level set of a smooth real-valued function on X .Let X be a connected finite open Riemann surface. Suppose the zero set L of a non-constantsmooth real valued function on X is regular. Each component of L is either a simple closedcurve or it can be parameterized by a continuous mapping (cid:96) : ( −∞ , ∞ ) → X . We call acomponent of the latter kind a simple relatively closed arc in X .A relatively closed curve γ in a connected finite open Riemann surface X is said to becontractible to a hole of X , if the following holds. Consider X as domain X c \ ∪C j on a closedRiemann surface X c . Here the C j are the holes, each is either a closed disc or a point. Thecondition is the following. For each pair U , U of open subsets of X c , ∪C j ⊂ U (cid:98) U , thereexists a homotopy of γ that fixes γ ∩ U and moves γ into U . Taking for U small enough BURGLIND J ¨ORICKE neighbourhoods of ∪C j we see that the homotopy moves γ into an annulus adjacent to one ofthe holes.For each relatively compact domain X (cid:48) (cid:98) X in X there is a finite cover of L ∩ X (cid:48) byopen subsets U k of X such that each L ∩ U k is connected. Each set L ∩ U k is contained ina component of L . Hence, only finitely many connected components of L intersect X (cid:48) . Let L be a connected component of L which is a simple relatively closed arc parameterized by (cid:96) : R → X . Since each set L ∩ U k is connected it is the image of an interval under (cid:96) . Takereal numbers t − and t +0 such that all these intervals are contained in ( t − , t +0 ). Then the images( (cid:96) (cid:0) ( −∞ , t − ) (cid:1) and ( (cid:96) (cid:0) ( t +0 , + ∞ ) (cid:1) are contained in X \ X (cid:48) , maybe, in different components. Suchparameters t − and t +0 can be found for each relatively compact deformation retract X (cid:48) of X .Hence for each relatively closed arc L ⊂ L the set of limit points L +0 of (cid:96) ( t ) for t → ∞ iscontained in a boundary component of X . Also, the set of limit points L − of (cid:96) ( t ) for t → −∞ is contained in a boundary component of X . The boundary components may be equal ordifferent. Moreover, if X (cid:48) (cid:98) X is a relatively compact domain in X which is a deformationretract of X , and a connected component L of L does not intersect X (cid:48) then L is contractibleto a hole of X . Indeed, X \ X (cid:48) is the union of disjoint annuli, each of which is adjacent to aboundary component of X , and the union of the limit sets L +0 ∪ L − is contained in a singleboundary component of X .Further, denote by L (cid:48) the union of all connected components of L that are simple relativelyclosed arcs. Consider those components L j of L (cid:48) that intersect X (cid:48) . There are finitely manysuch L j . Parameterize each L j by a mapping (cid:96) j : R → X . For each j we let [ t − j , t + j ] be acompact interval for which (cid:96) j ( R \ [ t − j , t + j ]) ⊂ X \ X (cid:48) . (4)Let X (cid:48)(cid:48) , X (cid:48) (cid:98) X (cid:48)(cid:48) (cid:98) X , be a domain which is a deformation retract of X such that (cid:96) j ([ t − j , t + j ]) ⊂X (cid:48)(cid:48) for each j . Then all connected components of L (cid:48) ∩ X (cid:48)(cid:48) , that do not contain a set (cid:96) j ([ t − j , t + j ]),are contractible to a hole of X (cid:48)(cid:48) . Indeed, each such component is contained in the union ofannuli X (cid:48)(cid:48) \ X (cid:48) . Some remarks on coverings.
Let X be a finite open Riemann surface with base point q andlet P : ˜ X → X be the universal covering map. By a covering P : Y → X we mean a continuousmap P from a topological space X to a topological space Y such that for each point x ∈ X thereis a neighbourhood V ( x ) of x such that the mapping P maps each connected component ofthe preimage of V ( x ) homeomorphically onto V ( x ). (Note that in function theory sometimesthese objects are called unlimited unramified coverings to reserve the notion ”covering” formore general objects.) Recall that a homeomorphism ϕ : ˜ X → ˜ X for which P ◦ ϕ = P iscalled a covering transformation (or deck transformation). The covering transformations forma group, denoted by Deck( ˜ X, X ). For each pair of points ˜ x , ˜ x ∈ ˜ X with P (˜ x ) = P (˜ x ) thereexists exactly one covering transformation that maps ˜ x to ˜ x . (See e.g. [8]).Throughout the paper we will fix a base point q ∈ X and a base point ˜ q ∈ P − ( q ) ⊂ ˜ X .The group of covering transformations of ˜ X can be identified with the fundamental group π ( X, q ) of X by the following correspondence. (See e.g. [8]).Take a covering transformation σ ∈ Deck( ˜
X, X ). Let ˜ γ be an arc in ˜ X with initial point σ (˜ q ) and terminating point ˜ q . Denote by Is ˜ q ( σ ) the element of π ( X, q ) represented by theloop P (˜ γ ). The mapping Deck( ˜ X, X ) (cid:51) σ → Is ˜ q ( σ ) ∈ π ( X, q ) is a group homomorphism.The homomorphism Is ˜ q is injective and surjective, hence it is a group isomorphism. Theinverse (Is ˜ q ) − is obtained as follows. Represent an element e ∈ π ( X, q ) by a loop γ .Consider the lift ˜ γ of γ to ˜ X that has initial point ˜ q . Then (Is ˜ q ) − ( e ) is the coveringtransformation that maps the endpoint of ˜ γ to ˜ q .For another point ˜ q of ˜ X and the point q def = P (˜ q ) ∈ X the isomorphism Is ˜ q : Deck( ˜ X, X )) → π ( X, q ) assigns to each σ ∈ Deck( ˜
X, X ) the element of π ( X, q ) that is represented by P (˜ γ ) IEMANN SURFACES OF SECOND KIND AND EFFECTIVE FINITENESS THEOREMS 9 for a curve ˜ γ in ˜ X that joins σ (˜ q ) with ˜ q . Is ˜ q is related to Is ˜ q as follows. Let ˜ α be an arc in˜ X with initial point ˜ q and terminating point ˜ q . Put q = P (˜ q ) and α = P ( ˜ α ). Then for theisomorphism Is α : π ( X, q ) → π ( X, q ) the equationIs ˜ q ( σ ) = Is α ◦ Is ˜ q ( σ ) , σ ∈ Deck( ˜
X, X ) , (5)holds, i.e. the following diagram Figure 2 is commutative. Indeed, let ˜ α − denote the curve π ( X, q ) Is α (cid:15) (cid:15) Deck( ˜
X, X ) Is ˜ q (cid:55) (cid:55) Is ˜ q (cid:39) (cid:39) π ( X, q ) Figure 2 that is obtained from ˜ α by inverting the direction on ˜ α , i.e. moving from ˜ q to ˜ q . The curve σ (( ˜ α ) − ) has initial point σ (˜ q ) and terminating point σ (˜ q ). Hence, for a curve ˜ γ in ˜ X thatjoins σ (˜ q ) with ˜ q , the curve σ (( ˜ α ) − ) ˜ γ ˜ α in ˜ X has initial point σ (˜ q ) and terminating point˜ q . Therefore P ( σ (( ˜ α ) − ) ˜ γ ˜ α ) represents Is ˜ q ( σ ). On the other hand P ( σ ( ˜ α − ) ˜ γ ˜ α ) = P ( σ ( ˜ α − )) P (˜ γ ) P ( ˜ α ) = α − γ α (6)represents Is α ( e ). In particular, if ˜ q (cid:48) ∈ P − ( q ) is another preimage of the base point q under the projection P , then the associated isomorphisms to the fundamental group π ( X, q )are conjugate, i.e. Is ˜ q (cid:48) ( e ) = ( e (cid:48) ) − Is ˜ q ( e ) e (cid:48) for each e ∈ π ( X, q ). The element e (cid:48) isrepresented by the projection of an arc in ˜ X with initial point ˜ q and terminating point ˜ q (cid:48) .Keeping fixed ˜ q qnd q we will say that a point ˜ q ∈ ˜ X and a curve α in X are compatibleif the diagram Figure 2 is commutative, equivalently, if equation (5) holds. We may also startwith choosing a curve α in X with initial point q and terminating point q . Then there is apoint ˜ q = ˜ q ( α ), such that ˜ q and α are compatible. Indeed, let ˜ α be the lift of α , that has initialpoint ˜ q . Denote the terminating point of ˜ α by ˜ q ( α ), and repeat the previous arguments.Let N be a subgroup of π ( X, q ). Denote by X ( N ) the quotient ˜ X (cid:30) (Is ˜ q ) − ( N ). Weobtain a covering ω N Id : ˜ X → X ( N ) with group of covering transformations isomorphic to N .The fundamental group of X ( N ) with base point ( q ) N def = ω N Id (˜ q ) can be identified with N .If N and N are subgroups of π ( X, q ) and N is a subgroup of N (we write N ≤ N ) thenthere is a covering map ω N N : ˜ X (cid:30) (Is ˜ q ) − ( N ) → ˜ X (cid:30) (Is ˜ q ) − ( N ), such that ω N N ◦ ω N Id = ω N Id .Moreover, the diagram Figure 3 below is commutative. ˜ X ω N Id X ( N ) ω N Id X ( N ) ω N N ω π ( X,q ) N ω π ( X,q ) N XP Figure 3
Indeed, take any point x ∈ ˜ X (cid:30) (Is ˜ q ) − ( N ) and a preimage ˜ x of x under ω N Id . There existsa neighbourhood V (˜ x ) of ˜ x in ˜ X such that V (˜ x ) ∩ σ ( V (˜ x )) = ∅ for all covering transformations σ ∈ Deck( ˜
X, X ). Then for j = 1 , ω N j , ˜ x Id def = ω N j Id | V (˜ x ) is a homeomorphismfrom V (˜ x ) onto its image denoted by V j . Put x = ω N j , ˜ x Id (˜ x ). The set V j ⊂ ˜ X (cid:30) (Is ˜ q ) − ( N j ) isa neighbourhood of x j for j = 1 , x (cid:48) ∈ ( ω N Id ) − ( x ) there is a covering transformation ϕ ˜ x, ˜ x (cid:48) in (Is ˜ q ) − ( N )which maps the neighbourhood V (˜ x (cid:48) ) of ˜ x (cid:48) conformally onto the neighbourhood V (˜ x ) of ˜ x so that on V (˜ x (cid:48) ) the equality ω N , ˜ x (cid:48) Id = ω N , ˜ x Id ◦ ϕ ˜ x, ˜ x (cid:48) holds. Choose ˜ x ∈ ( ω N Id ) − ( x ) anddefine ω N N ( y ) = ω N , ˜ xN ( y ) def = ω N , ˜ x Id (( ω N , ˜ x Id ) − ( y )) for each y ∈ V . We get a correctly definedmapping from V onto V . Indeed, since N is a subgroup of N , the covering transformation ϕ ˜ x, ˜ x (cid:48) is contained in (Is ˜ q ) − ( N ), and we get for another point ˜ x (cid:48) ∈ ( ω N Id ) − ( x ) the equality ω N , ˜ x (cid:48) Id = ω N , ˜ x Id ◦ ϕ ˜ x, ˜ x (cid:48) . Hence, for y ∈ V ( x ) ω N , ˜ x (cid:48) Id ◦ ( ω N , ˜ x (cid:48) Id ) − ( y ) = ( ω N , ˜ x Id ◦ ϕ ˜ x, ˜ x (cid:48) ) ◦ ( ω N , ˜ x Id ◦ ϕ ˜ x, ˜ x (cid:48) ) − ( y ) = ω N , ˜ x Id ◦ ( ω N , ˜ x Id ) − ( y ) . Since the mappings ω N j , ˜ x Id , j = 1 , , are homeomorphisms from V (˜ x ) onto its image, the map-ping ω N N is a homeomorphism from V ( x ) onto V ( x ). The same holds for all preimages of V ( x ) under ω N N . Hence, ω N N is a covering map. The commutativity of the part of the diagramthat involves the mappings ω N Id , ω N Id , and ω N N is clear by construction.The existence of ω π ( X,q ) N and the equality P = ω π ( X,q ) N ◦ ω N Id follows by applying the abovearguments with N = π ( X, q ). The equality P = ω π ( X,q ) N ◦ ω N Id follows in the same way.Since P = ω π ( X,q ) N ◦ ω N N ◦ ω N Id = ω π ( X,q ) N ◦ ω N Id , we have ω π ( X,q ) N ◦ ω N N = ω π ( X,q ) N We will also use the notation ω N def = ω N Id and ω N def = ω π ( X,q ) N for a subgroup N of π ( X, q ).In our situation N will usually be a subgroup (cid:104) ˜ E(cid:105) of π ( X, q ) generated by a system ˜ E ofelements of E ⊂ π ( X, q ).Consider a set E (cid:48) ⊂ π ( X, x ) that consists of 2 g (cid:48) + m (cid:48) elements of E , such that among themare exactly g (cid:48) of the chosen pairs in E that correspond to g (cid:48) of the handles of X , and from allremaining pairs of elements of E corresponding to a handle at most one element is containedin E (cid:48) . Then X ( (cid:104)E (cid:48) (cid:105) ) is a surface of genus g (cid:48) with m (cid:48) holes. Indeed, consider the chosen loops α j , β j , and γ g + (cid:96) , that represent the elements of E . The union of those of them that representelements of E (cid:48) can be identified with a deformation retract of X ( (cid:104)E (cid:48) (cid:105) ). To see this, we lift thecollection of all α j , β j , and γ j , representing elements of E to arcs in the universal covering ˜ X of X with initial point ˜ q , and take the quotient X ( (cid:104)E (cid:48) (cid:105) ) of ˜ X by the action of Is ˜ q ( (cid:104)E (cid:48) (cid:105) ). Thelifts of the loops representing elements of E \ E (cid:48) project to arcs in X ( (cid:104)E (cid:48) (cid:105) ). The lifts of theloops representing elements of E (cid:48) project to loops in X ( (cid:104)E (cid:48) (cid:105) ), the union of which is a bouquetof circles which is a deformation retract of X ( (cid:104)E (cid:48) (cid:105) ).Let again N ≤ N be subgroups of π ( X, q ). Consider the covering ω N N : ˜ X (cid:30) (Is ˜ q ) − ( N ) → ˜ X (cid:30) (Is ˜ q ) − ( N ). Let β be a simple relatively closed curve in ˜ X (cid:30) (Is ˜ q ) − ( N ). Then ( ω N N ) − ( β )is the union of simple relatively closed curves in ˜ X (cid:30) (Is ˜ q ) − ( N ) and ω N N : ( ω N N ) − ( β ) → β isa covering. Indeed, we cover β by small discs U k in ˜ X (cid:30) (Is ˜ q ) − ( N ) such that for each k therestriction of ω N N to each connected component of ( ω N N ) − ( U k ) is a homeomorphism onto U k , IEMANN SURFACES OF SECOND KIND AND EFFECTIVE FINITENESS THEOREMS 11 and U k intersects β along a connected set. Take any k with U k ∩ β (cid:54) = ∅ . Consider the preimages( ω N N ) − ( U k ). Restrict ( ω N N ) to the intersection of each such preimage with ( ω N N ) − ( β ). Weobtain a homeomorphism onto U k ∩ β . The extremal length of monodromies.
Let as before X be a connected finite open Rie-mann surface with base point q , and ˜ q a point in the universal covering ˜ X for which P (˜ q ) = q for the covering map P : ˜ X → X .Recall that for an arbitrary point q ∈ X the free homotopy class of an element e of thefundamental group π ( X, q ) can be identified with the conjugacy of elements of π ( X, q ) con-taining e and is denoted by (cid:98) e . Notice that for e ∈ π ( X, q ) and a curve α in X with initialpoint q and terminating point q the free homotopy classes of e and of e = Is α ( e ) coincide,i.e. (cid:98) e = (cid:98) e . Consider a simple smooth relatively closed curve L in X . We will say that a freehomotopy class of curves (cid:98) e intersects L if each representative of (cid:98) e intersects L . Choose anorientation of L . The intersection number of (cid:98) e with the oriented curve L is the intersectionnumber with L of some (and, hence, of each) smooth loop representing (cid:98) e that intersects L transversally. This intersection number is the sum of the intersection numbers over all inter-section points. The intersection number at an intersection point equals +1 if the orientationdetermined by the tangent vector to L followed by the tangent vector to the curve representing (cid:98) e is the orientation of X , and equals − A be an annulus. A continuous mapping ω : A → X is said to represent a conjugacyclass (cid:98) e of elements of the fundamental group π ( X, q ) for a point q ∈ X , if for each simpleclosed curve γ in A that is homologous to a boundary component of A the restriction of themapping to γ represents (cid:98) e or the inverse (cid:100) e − .Let A be an annulus with base point p , and let ω be a continuous mapping from A to a finiteRiemann surface X with base point q such that ω ( p ) = q . We write ω : ( A, p ) → ( X, q ). Themapping is said to represent the element e of the fundamental group π ( X, q ) if the preimage( ω ∗ ) − ( e ) under the induced mapping ω ∗ : π ( A, p ) → π ( X, q ) between fundamental groups isa generator of π ( A, p ). There are two generators and they are inverse to each other.We associate to each element e ∈ π ( X, q ) of the free group π ( X, q ) the annulus X ( (cid:104) e (cid:105) ) =˜ X (cid:30) (Is ˜ q ) − ( (cid:104) e (cid:105) ) with base point q (cid:104) e (cid:105) = ω (cid:104) e (cid:105) Id (˜ q ) and the covering map ω (cid:104) e (cid:105) def = ω π ( X,q ) (cid:104) e (cid:105) : X ( (cid:104) e (cid:105) ) → X . Then ω (cid:104) e (cid:105) ( q (cid:104) e (cid:105) ) = q . The mapping ω π ( X,q ) (cid:104) e (cid:105) : ( X ( (cid:104) e (cid:105) ) , q (cid:104) e (cid:105) ) → ( X, q )represents e . Indeed, let γ be a closed curve in X with base point q which represents e .Consider the lift of γ to a curve ˜ γ in ˜ X with terminating point ˜ q . Its initial point equals(Is ˜ q ) − ( e )(˜ q ). Hence, the projection ω (cid:104) e (cid:105) (˜ γ ) of ˜ γ to ˜ X (cid:30) (Is ˜ q ) − ( (cid:104) e (cid:105) ) is a closed curvedenoted by γ (cid:104) e (cid:105) which is a generator of the fundamental group π ( X ( (cid:104) e (cid:105) ) , ( q ) (cid:104) e (cid:105) ). Since ω (cid:104) e (cid:105) ( γ (cid:104) e (cid:105) ) = ω (cid:104) e (cid:105) ( ω (cid:104) e (cid:105) (˜ γ )) = P (˜ γ ) = γ , we proved, that the locally conformal mapping ω (cid:104) e (cid:105) : (( X ( (cid:104) e (cid:105) ) , ( q ) (cid:104) e (cid:105) ) → ( X, q ) represents e .Take a curve α in X that joins q and q , and a point ˜ q = ˜ q ( α ) ∈ ˜ X such that α and ˜ q are compatible, i.e. Is ˜ q = Is α ◦ Is ˜ q (see equation (5)). Put e = Is α ( e ). By equation (5))(Is ˜ q ) − ( e ) = (Is ˜ q ) − ( e ), hence, ˜ X (cid:30) (Is ˜ q ) − ( (cid:104) e (cid:105) ) = ˜ X (cid:30) (Is ˜ q ) − ( (cid:104) e (cid:105) ) = X ( (cid:104) e (cid:105) ). The locallyconformal mapping ω (cid:104) e (cid:105) : X ( (cid:104) e (cid:105) ) → X takes the point q (cid:104) e (cid:105) def = ω (cid:104) e (cid:105) (˜ q ) to q ∈ X . Moreover, ω (cid:104) e (cid:105) : ( X ( (cid:104) e (cid:105) ) , q (cid:104) e (cid:105) ) → ( X, q ) represents e ∈ π ( X, q ). This can be seen by repeating theprevious arguments.Let α be an arbitrary curve in X joining q with q , and ˜ q ∈ P − ( q ) be arbitrary (i.e., α and ˜ q are not required to be compatible). Let e ∈ π ( X, q ). Denote the projection ˜ X → ˜ X (cid:30) (Is ˜ q ) − ( (cid:104) e (cid:105) ) by ω (cid:104) e (cid:105) , ˜ q , and the projection ˜ X (cid:30) (Is ˜ q ) − ( (cid:104) e (cid:105) ) → X by ω (cid:104) e (cid:105) , ˜ q . Put q (cid:104) e (cid:105) , ˜ q = ω (cid:104) e (cid:105) , ˜ q (˜ q ). Then the mapping ω (cid:104) e (cid:105) , ˜ q : ˜ X (cid:30) (cid:0) (Is ˜ q ) − ( (cid:104) e (cid:105) ) , q (cid:104) e (cid:105) , ˜ q (cid:1) → ( X, q ) represents e .The mapping ω (cid:104) e (cid:105) : X ( (cid:104) e (cid:105) ) → X represents the free homotopy class (cid:98) e . Moreover, X ( (cid:104) e (cid:105) )is the ”thickest” annulus which admits a holomorphic mapping to X , that represents theconjugacy class (cid:98) e . This means, that the annulus has smallest extremal length among annuli with the mentioned property. Indeed, let A ω −−→ X be a holomorphic mapping of an annulusthat represents (cid:98) e . A is conformally equivalent to a round annulus in the plane, hence, we mayassume that A has the form A = { z ∈ C : r < | z | < R } for positive real numbers r < R .Take a simple closed curve γ A in A that is homologous to a boundary component of A .After maybe, reorienting γ A , its image under ω represents the class (cid:98) e . Choose a point q A in γ A , and put q = ω ( q A ). Then γ A represents a generator of π ( A, q A ) and γ represents anelement e of π ( X, q ) in the conjugacy class (cid:98) e . Choose a curve α in X with initial point q and terminating point q , and a point ˜ q in ˜ X so that α and ˜ q are compatible, and, hence, for e = Is α ( e ) the equality (Is ˜ q ) − ( e ) = (Is ˜ q ) − ( e ) holds. Let L be the relatively closed arc( p · R ) ∩ A in A through p along the radius. After a homotopy of γ A with fixed base point,we may assume that its base point q A is the only point of γ A that is contained in L . Therestriction ω | ( A \ L ) lifts to a mapping ˜ ω : ( A \ L ) → ˜ X , that extends continuously to the twostrands L ± of L . (Here L − contains the initial point of γ A .) Let p ± be the copies of p on thetwo strands L ± . We choose the lift ˜ ω so that ˜ ω ( p + ) = ˜ q . Since the mapping ( A, q A ) → ( X, q )represents e , we obtain ˜ ω ( p − ) = σ (˜ q ) for σ = (Is ˜ q ) − ( e ). Then for each z ∈ L the coveringtransformation σ maps the point ˜ z + ∈ ˜ ω ( L + ) for which P (˜ z + ) = z to the point ˜ z − ∈ ˜ ω ( L − )for which P (˜ z − ) = z . Hence ω lifts to a holomorphic mapping ι : A → X ( (cid:104) e (cid:105) ). By Lemma 7of [18] λ ( A ) ≥ λ ( X ( (cid:104) e (cid:105) )).For each point q ∈ X and each element e ∈ π ( X, q ) we denote by A ( (cid:98) e ) the conformal classof the ”thickest” annulus that admits a holomorphic mapping into X that represents (cid:98) e . Foreach ˜ q (cid:48) ∈ ˜ X and each element e ∈ π ( X, P (˜ q (cid:48) )) the mapping ω (cid:104) e (cid:105) , ˜ q (cid:48) : ˜ X (cid:30) (Is ˜ q (cid:48) ) − ( (cid:104) e (cid:105) ) → X represents the conjugacy class (cid:98) e of e . Hence, if (cid:98) e = (cid:98) e , then λ ( ˜ X (cid:30) (Is ˜ q (cid:48) ) − ( (cid:104) e (cid:105) )) ≤ λ ( A ( (cid:98) e )).Interchanging the role of the representatives e and e of (cid:98) e and of the base points ˜ q (cid:48) and ˜ q of ˜ X , we see that in fact λ ( ˜ X (cid:30) (Is ˜ q (cid:48) ) − ( (cid:104) e (cid:105) )) = λ ( A ( (cid:98) e )) for each ˜ q (cid:48) ∈ ˜ X and each element e ∈ π ( X, P (˜ q (cid:48) )). Notice that A ( (cid:100) e − ) = A ( (cid:98) e ).The following lemma will be crucial for the estimate of the L − -invariant of the monodromiesof holomorphic mappings from a finite open Riemann surface to C \ {− , } . Lemma 1.
Let f : X → C \ {− , } be a non-contractible holomorphic function on a connectedfinite open Riemann surface X , such that is a regular value of Im f . Assume that L is asimple relatively closed curve in X such that f ( L ) ⊂ ( − , . Let q ∈ L and q (cid:48) = f ( q ) .If for an element e ∈ π ( X, q ) the free homotopy class (cid:98) e intersects L , then either the reducedword f ∗ ( e ) ∈ π ( C \{− , } , q (cid:48) ) is a non-zero power of a standard generator of π ( C \{− , } , q (cid:48) ) or the inequality L − ( f ∗ ( e )) ≤ πλ ( A ( (cid:98) e )) (7) holds. Notice that we make a normalization in the statement of the Lemma by requiring that f maps L into the interval ( − , R \ {− , } .Lemma 1 will be a consequence of the following lemma. Lemma 2.
Let f , L , q ∈ L be as in Lemma 1, and e ∈ π ( X, q ) . Let ˜ q be an arbitrarypoint in P − ( q ) . Consider the annulus A def = ˜ X (cid:30) (Is ˜ q ) − ( (cid:104) e (cid:105) ) and the holomorphic projection ω A def = ω (cid:104) e (cid:105) , ˜ q . Put q A def = ω (cid:104) e (cid:105) , ˜ q (˜ q ) and let L A ⊂ A be the connected component of ( ω A ) − ( L ) that contains q A . Then the mapping ω A : ( A, q A ) → ( X, q ) represents e .If (cid:98) e intersects L , then L A is a relatively closed curve in A that has limit points on bothboundary components of A , and the lift f ◦ ω A is a holomorphic function on A that maps L A into ( − , . Proof of Lemma 2.
Let γ : [0 , → X be a curve with base point q in X that represents e ,and let ˜ γ be the lift of γ to ˜ X with terminating point ˜ γ (1) equal to ˜ q . Put σ def = (Is ˜ q ) − ( e ).Then the initial point ˜ γ (0) equals σ (˜ q ). IEMANN SURFACES OF SECOND KIND AND EFFECTIVE FINITENESS THEOREMS 13
All connected components of P − ( L ) are relatively closed curves in ˜ X ∼ = C + (where C + denotes the upper half-plane) with limit points on the boundary of ˜ X . Indeed, the lift f ◦ P of f to ˜ X takes values in ( − ,
1) on P − ( L ). Hence, by the maximum principle applied toexp( f ◦ P ) the latter cannot contain compact connected components since f is not constant.Let ˜ L ˜ q be the connected component of P − ( L ) that contains ˜ q . The point σ (˜ q ) cannot becontained in ˜ L ˜ q . Indeed, otherwise there exists a curve in ˜ X that is homotopic to ˜ γ with fixedendpoints and is contained in ˜ L ˜ q . A small translation of the latter curve gives a curve in ˜ X that does not intersect P − ( L ) and projects under P to a curve that is free homotopic to γ .This contradicts the fact that (cid:98) e intersects L .Each of the two connected components ˜ L ˜ q and σ ( ˜ L ˜ q ) divides ˜ X . Let Ω be the domain on˜ X that is bounded by ˜ L ˜ q and σ ( ˜ L ˜ q ) and parts of the boundary of ˜ X . After a homotopy of˜ γ that fixes the endpoints we may assume that ˜ γ ((0 , Ω . Indeed, for eachconnected component of ˜ γ ((0 , \ Ω there is a homotopy with fixed endpoints that moves theconnected component to an arc on ˜ L ˜ q or σ ( ˜ L ˜ q ). A small perturbation yields a curve ˜ γ (cid:48) whichis homotopic with fixed endpoints to ˜ γ and has interior contained in Ω . Notice that ˜ γ (cid:48) ((0 , σ k ( ˜ L ˜ q ).The curve ω (cid:104) e (cid:105) , ˜ q (˜ γ (cid:48) ) is a closed curve on A that represents a generator of the fundamentalgroup of A with base point q A . Moreover, ω A ◦ ω (cid:104) e (cid:105) , ˜ q (˜ γ (cid:48) ) = ω (cid:104) e (cid:105) , ˜ q ◦ ω (cid:104) e (cid:105) , ˜ q (˜ γ (cid:48) ) = P (˜ γ (cid:48) ) represents e . Hence, the mapping ω A : ( A, q A ) → ( X, q ) represents e .The curve ω (cid:104) e (cid:105) , ˜ q (˜ γ (cid:48) ) intersects L (cid:104) e (cid:105) = ω (cid:104) e (cid:105) , ˜ q ( ˜ L ˜ q ) exactly once. Hence, L (cid:104) e (cid:105) has limit pointson both boundary circles of A for otherwise L (cid:104) e (cid:105) would intersect one of the components of A \ ω (cid:104) e (cid:105) , ˜ q (˜ γ (cid:48) ) along a set which is relatively compact in A , and ˜ γ (cid:48) would have intersectionnumber zero with L (cid:104) e (cid:105) . It is clear that f ◦ ω A ( L A ) = f ( L ) ⊂ ( − , (cid:50) Proof of Lemma 1.
Let ω A : ( A, q A ) → ( X, q ) be the holomorphic mapping from lemma2 that represents e , and let L A (cid:51) q A be the relatively closed curve in A with limit set onboth boundary components of A . Consider a loop γ A : [0 , → A with base point γ (0) = γ (1) = q A such that γ A ((0 , ⊂ A \ L A . Then γ = ω A ( γ A ) represents e . The mapping f ◦ ω A is holomorphic on A and f ◦ ω A ( γ A ) = f ( γ ) represents f ∗ ( e ) ∈ π ( C \ {− , } , q (cid:48) ) with q (cid:48) = f ◦ ω A ( q A ) = f ( q ) ∈ ( − , f ◦ ω A ( γ A ) also represents the element ( f ∗ ( e )) tr ∈ π tr ( C \{− , } ) in the relative fundamental group fundamental group π ( C \{− , } , ( − , π tr ( C \ {− , } ) corresponding to f ∗ ( e ).We prove now that Λ tr ( f ∗ ( e )) ≤ λ ( A ). Let A (cid:98) A be any relatively compact annulus in A with smooth boundary such that q A ∈ A . If A is sufficiently large, then the connectedcomponent L A of L A ∩ A that contains q A has endpoints on different boundary componentsof A . The set A \ L A is a curvilinear rectangle. The horizontal curvilinear sides are thestrands of the cut that are reachable from the curvilinear rectangle moving counterclockwise,or clockwise, respectively. The vertical curvilinear sides are obtained from the boundary circlesof A by removing an endpoint of the arc L A . Since f ◦ ω A maps L A to ( − , f ◦ ω A to A \ L A represents ( f ∗ ( e )) tr . Hence, Λ tr ( f ∗ ( e )) ≤ λ ( A \ L A ) . (8)Moreover, λ ( A \ L A ) ≤ λ ( A ) . (9)This is a consequence of the following facts. First, λ ( A \ L A ) is equal to the extremal length λ ( Γ ( A \ L A )) in the sense of Ahlfors [1] of the family Γ ( A \ L A ) of curves in the curvilinearrectangle A \ L A that join the two horizontal sides of the curvilinear rectangle. Further, λ ( A ) is equal to the extremal length λ ( Γ ( A )) [1] of the family Γ ( A ) of curves in A thatare free homotopic to representatives of the positive generator of A . Finally, by [1], Ch.1 Theorem 2, the inequality λ ( Γ ( A \ L A )) ≤ λ ( Γ ( A )) (10)holds. We obtain the inequality Λ tr ( f ∗ ( e )) ≤ λ ( A ) for each annulus A (cid:98) A , hence, since A belongs to the class A ( (cid:98) e ) of conformally equivalent annuli, Λ tr ( f ∗ ( e )) ≤ λ ( A ( (cid:98) e )) , (11)and the Lemma follows from Theorem F. (cid:50) The monodromies along two generators.
In the following Lemma we combine the in-formation on the monodromies along two generators of the fundamental group π ( X, q ). Weallow the situation when the monodromy along one generator or along both generators of thefundamental group of X is a power of a standard generator of π ( C \ {− , } , f ( q )). Lemma 3.
Let f : X → C \ {− , } be a holomorphic function on a connected open Riemannsurface X such that is a regular value of the imaginary part of f . Suppose f maps a simplerelatively closed curve L in X to ( − , , and q is a point in L . Let e (1) and e (2) be primitiveelements of π ( X, q ) . Suppose that for each e = e (1) , e = e (2) , and e = e (1) e (2) , the freehomotopy class (cid:98) e intersects L . Then either f ∗ ( e ( j ) ) , j = 1 , , are (trivial or non-trivial)powers of the same standard generator of π ( C \ {− , } , q (cid:48) ) with q (cid:48) = f ( q ) ∈ ( − , , or eachof them is the product of at most two elements w and w of π ( C \ {− , } , q (cid:48) ) with L − ( w j ) ≤ πλ e (1) ,e (2) , j = 1 , , where λ e (1) ,e (2) def = max { λ ( A ( (cid:100) e (1) )) , λ ( A ( (cid:100) e (2) )) , λ ( A ( (cid:92) e (1) e (2) )) } . Hence, L − ( f ∗ ( e ( j ) ) ≤ πλ e (1) ,e (2) , j = 1 , . (12) Proof.
If the monodromies f ∗ ( e (1) ) and f ∗ ( e (2) ) are not powers of a single standard generator(the identity is considered as zeroth power of a standard generator) we obtain the following. Atmost two of the elements, f ∗ ( e (1) ), f ∗ ( e (2) ), and f ∗ ( e (1) e (2) ) = f ∗ ( e (1) ) f ∗ ( e (2) ), are powers of astandard generator, and if two of them are powers of a standard generator, then they are non-zero powers of different standard generators. If two of them are non-zero powers of standardgenerators, then the third has the form a k(cid:96) a k (cid:48) (cid:96) (cid:48) with a (cid:96) and a (cid:96) (cid:48) being different generators and k and k (cid:48) being non-zero integers. By Lemma 1 the L − of the third element does not exceed2 πλ e (1) ,e (2) . On the other hand it equals log(3 | k (cid:48) | ) + log(3 | k | ). Hence, L − ( a k(cid:96) ) = log(3 | k | ) ≤ πλ e (1) ,e (2) and L − ( a k (cid:48) (cid:96) (cid:48) ) = log(3 | k (cid:48) | ) ≤ πλ e (1) ,e (2) .If two of the elements f ∗ ( e (1) ), f ∗ ( e (2) ), and f ∗ ( e (1) e (2) ) = f ∗ ( e (1) ) f ∗ ( e (2) ), are not powersof a standard generator, then the L − of each of the two elements does not exceed 2 πλ e (1) ,e (2) .Since the L − of an element coincides with the L − of its inverse, the third element is theproduct of two elements with L − not exceeding 2 πλ e (1) ,e (2) . Since for x, x (cid:48) ≥ x + x (cid:48) ) ≤ log x + log x (cid:48) holds, the L − of the product does not exceed the sum of the L − ofthe factors. Hence the L − of the third element does not exceed 4 πλ e (1) ,e (2) . Hence, inequality(12) holds. (cid:50) The following proposition states the existence of suitable connected components of the zeroset of the imaginary part of certain analytic functions on tori with a hole and on planardomains. For any subset E (cid:48) of π ( X ; q ) we denote by ( E (cid:48) ) − the set of all elements that areinverse to elements in E (cid:48) . Recall that E j is the set of primitive elements of π ( X, q ) which canbe written as product of at most j elements of E ∪ ( E ) − for the set E of generators of π ( X, q )chosen in the introduction. IEMANN SURFACES OF SECOND KIND AND EFFECTIVE FINITENESS THEOREMS 15
Proposition 3.
Let X be a torus with a hole or a planar domain with base point q andfundamental group π ( X, q ) , and let E be the set of generators of π ( X, q ) that was chosenin Section 1. Let f : X → C \ {− , } be a non-contractible holomorphic mapping such that is a regular value of Im f . Then there exist a simple relatively closed curve L ⊂ X suchthat f ( L ) ⊂ R \ {− , } , and a set E (cid:48) ⊂ E ⊂ π ( X, q ) of primitive elements of π ( X, q ) ,such that the following holds. Each element e j, ∈ E ⊂ π ( X, q ) is the product of at most twoelements of E (cid:48) ∪ ( E (cid:48) ) − . Moreover, for each e ∈ π ( X, q ) which is the product of one or twoelements from E (cid:48) the free homotopy class (cid:98) e has positive intersection number with L (aftersuitable orientation of L ). Notice the following fact. If f is irreducible, then it is not contractible, and, hence, thepreimage f − ( R ) is not empty.Denote by M a M¨obius transformation which permutes the points − , , ∞ and maps theinterval ( −∞ , −
1) onto ( − , M be a M¨obius transformation which permutes thepoints − , , ∞ and maps the interval (1 , ∞ ) onto ( − , M def = Id.The main step for the proof of Theorem 1 is the following Proposition 4.Recall that λ j ( X ) was defined in the introduction. Since for e ∈ π ( X, q ) the equality λ ( ˜ X (cid:30) (Is ˜ q ) − ( (cid:104) e (cid:105) )) = λ ( A ( (cid:98) e )), λ j ( X ) is the maximum over e ∈ E j of λ ( A ( (cid:98) e )). Proposition 4.
Let X be a connected finite open Riemann surface with base point q , and let E be the set of generators of π ( X, q ) that was chosen in Section 1. Suppose f : X → C \ {− , } is an irreducible holomorphic mapping, such that is a regular value of Im f . Then for one ofthe functions M l ◦ f, l = 0 , , , which we denote by F , there exists a point q ∈ X (dependingon f ), such that the point q (cid:48) def = F ( q ) is contained in ( − , , and a curve α in X joining q with q , such that the following holds. For each element e j ∈ Is α ( E ) the monodromy F ∗ ( e j ) isthe product of at most four elements of π (( C ) \ {− , } , q (cid:48) ) of L − not exceeding πλ ( X ) and,hence, L − ( F ∗ ( e j )) ≤ πλ ( X ) for each j. (13)Notice that F is irreducible iff f is irreducible. By Theorem E and Lemma 4 of [21] the non-contractible mapping F is irreducible if and only if the image F ∗ ( π ( X, q )) of the monodromyhomomorohism is not generated by a single element of the fundamental group π ( C \{− , } , q (cid:48) ), q (cid:48) ∈ ( − , π ( C \ {− , } , q (cid:48) ).Notice, that all monodromies of contractible mappings are equal to zero, hence the inequality(13) holds automatically for contractible mappings.We postpone the proof of the two propositions and prove first the Theorem 1. Proof of Theorem 1.
Let X be a connected finite open Riemann surface (possibly of secondkind) with base point q . Using Theorem E and Proposition 4 we want to bound the number ofirreducible free homotoy classes of mappings from X to C \ {− , } that contain a holomorphicmapping. Consider a free homotopy class that is represented by an irreducible holomorphicmapping f : X → C \ {− , } . Recall that irreducible mappings are not contractible. Con-sider an arbitrary open Riemann surface X (cid:98) X which is relatively compact in X and isa deformation retract of X . Let ε be a small enough real number ε , such that the function( f − iε ) | X takes values in C \ {− , } and 0 is a regular value of its imaginary part. Put f = ( f − iε ) | X . Notice that the irreducible mapping f on X is free homotopic to f | X .We identify the fundamental groups of X and of X by the inclusion mapping from X to X .The statement of Proposition 4 applies to the function f on X . Associate to f the function F = M l ◦ f on X , and the points q and q (cid:48) from Proposition 4. The proposition providesinformation about the monodromies of the mapping F along the elements Is α ( E ) for somecurve α that was chosen depending on f , and has initial point q and terminating point q . Wehave to relate this information to the monodromies of the mapping F along the elements of thesystem E of generators of the fundamental group π ( X, q ) ∼ = π ( X , q ) with base point q , that was chosen in Section 1 and is independent on the function f . Write e j = Is α ( e j, ) ∈ π ( X, q )for e j, ∈ π ( X, q ). The image of α under the mapping F is the curve β = F ◦ α in C \ {− , } with initial point F ( q ) and terminating point q (cid:48) . We have F ∗ ( e j, ) = (Is β ) − ( F ∗ ( e j )). Choosea homotopy F t , t ∈ [0 , F def = F with a (smooth) mapping F denoted by ˜ F , so that the value F t ( q ) moves from the point F ( q ) to q (cid:48) along the curve β .Then ˜ F ( q ) = q (cid:48) and ˜ F ∗ ( e j, ) = F ∗ ( e j ).Inequality (13) gives for each j the inequality L − ( ˜ F ∗ ( e j, )) ≤ πλ ( X ) . (14)Notice that if f is a contractible function, we may associate to it the function ˜ F on X whichis equal f | X and the inequality (14) is automatically satisfied for the monodromies of ˜ F .By Lemma 1 of [20] there are at most e πλ ( X ) + 1 ≤ e πλ ( X ) different reduced words w ∈ π ( C \ {− , } ) ,
0) (including the identity) with L − ( w ) ≤ πλ ( X ). Identify standardgenerators of π ( C \ {− , } , q (cid:48) ) with standard generators of π ( C \ {− , } ,
0) by the canonicalisomorphism. We obtain, that ˜ F ∗ is contained in a set of at most ( e πλ ( X ) ) g + m differenthomomorphisms π ( X , q ) ∼ = π ( X, q ) → π ( C \{− , } , q (cid:48) ) ∼ = π ( C \{− , } , X to C \ {− , } and conjugacy classes of homomorphisms π ( X , q ) → π ( C \ {− , } , q (cid:48) ) ∼ = π ( C \ {− , } , F is free homotopic to ˜ F , the function F represents one of at most( e πλ ( X ) ) g + m different free homotopy classes from X to the twice punctured complexplane.The function f is equal to M − l ◦ F for one of the numbers 0, 1, or 2, hence it representsone of 3( e πλ ( X ) ) g + m free homotopy classes of mappings from X to the twice puncturedcomplex plane. Since the considered set of mappings f on X is equal to the set of mappings( f − iε ) | X for a holomorphic map f on X , there are no more than 3( e πλ ( X ) ) g + m freehomotopy classes of irreducible mappings X → C \ {− , } containing a holomorphic mapping f . Theorem 1 is proved with the upper bound 3( e πλ ( X ) ) g + m for an arbitrary relativelycompact domain X ⊂ X that is a deformation retract of X .It remains to prove that λ ( X ) = inf { λ ( X ) : X (cid:98) X is a deformation retract of X } .We have to prove that for each e ∈ π ( X, x ) the number λ ( A ( (cid:98) e )) = λ ( (cid:101) X (cid:30) (Is ˜ q ) − ( (cid:104) e (cid:105) ))is equal to the infimum of λ ( (cid:102) X (cid:30) (Is ˜ q ) − ( (cid:104) e (cid:105) )) over all X being open relatively compactsubsets of X which are deformation retracts of X . Here (cid:102) X is the universal covering of X ,and the fundamental groups of X and X are identified. (cid:102) X ( (cid:101) X , respectively) can be defined asset of homotopy classes of arcs in X (in X , respectively) joining q with a point q ∈ X (in X respectively) equipped with the complex structure induced by the projection to the endpointof the arcs, and the point ˜ q corresponds to the class of the constant curve. The isomorphism(Is ˜ q ) − from π ( X , q ) to the group of covering transformations on (cid:102) X is defined in the sameway as it was done for X instead of X . We see that there is a holomorphic mapping from (cid:101) X (cid:30) (Is ˜ q ) − ( (cid:104) e (cid:105) ) into (cid:101) X (cid:30) (Is ˜ q ) − ( (cid:104) e (cid:105) ). Hence, the extremal length of the first set is notsmaller than the extremal length of the second set.Vice versa, take any annulus A which is a relatively compact subset of A ( (cid:98) e ) and is adeformation retract of A ( (cid:98) e ). Its projection to X is relatively compact in X , hence, it iscontained in a relatively compact deformation retract X of X . Hence, A can be consideredas subset of (cid:102) X (cid:30) (Is ˜ q ) − ( (cid:104) e (cid:105) ), and, hence, λ ( (cid:102) X (cid:30) (Is ˜ q ) − ( (cid:104) e (cid:105) )) ≤ λ ( A ). Since λ ( A ( (cid:98) e )) =inf { λ ( A ) : A (cid:98) A ( (cid:98) e ) is a deformation retract of A ( (cid:98) e ) } we are done. (cid:50) We proved a slightly stronger statement, namely, the number of homotopy classes of map-pings X → C \ {− , } that contain a contractible holomorphic mapping or an irreducibleholomorphic mapping does not exceed ( e πλ ( X ) ) g + m . IEMANN SURFACES OF SECOND KIND AND EFFECTIVE FINITENESS THEOREMS 17
Proof of Proposition 3.
Denote the zero set { x ∈ X : Im f ( x ) = 0 } by L . Since f is notcontractible, L (cid:54) = ∅ .
1. A torus with a hole.
Assume first that X is a torus with a hole with base point q . Fornotational convenience we denote by e and e (cid:48) the two elements of the set of generators E of π ( X, q ) that was chosen in Section 1. We claim that there is a connected component L of L which has (after suitable orientation) positive intersection number with the free homotopyclass of one of the elements of E , say with (cid:98) e , and L has positive intersection number withone of the classes (cid:98) e (cid:48) , or (cid:92) ( e (cid:48) ) − , or (cid:100) e e (cid:48) .The claim is easy to prove in the case when there is a component of L which is a simpleclosed curve that is not contractible and not contractible to the hole of X . Indeed, considerthe inclusion of X into a closed torus X c and the homomorphism on fundamental groups π ( X, q ) → π ( X c , q ) induced by the inclusion. Denote by e c and e (cid:48) c the images of e and e (cid:48) under this homomorphism. Notice that e c and e (cid:48) c commute. The (image under theinclusion of the) curve L is a simple closed non-contractible curve in X c . It represents thefree homotopy class of an element ( e c ) j ( e (cid:48) c ) k for some integers j and k which are not bothequal to zero. Hence, L is not null-homologous in X c , and by the Poincar´e Duality Theoremfor one of the generators, say for e c , the representatives of the free homotopy class (cid:98) e c havenon-zero intersection number with L . After suitable orientation of L , we may assume thatthis intersection number is positive. There is a representative of the class (cid:98) e c which is containedin X , hence, (cid:98) e has positive intersection number with L .Suppose all compact connected components of L are contractible or contractible to the holeof X . Consider a relatively compact domain X (cid:48)(cid:48) (cid:98) X in X with smooth boundary which is adeformation retract of X such that for each connected component of L at most one componentof its intersection with X (cid:48)(cid:48) is not contractible to the hole of X (cid:48)(cid:48) . (See the paragraph on ”Regularzero sets”.) There is at least one component of L ∩ X (cid:48)(cid:48) that is not contractible to the holeof X (cid:48)(cid:48) . Indeed, otherwise the free homotopy class of each element of E could be representedby a loop avoiding L , and, hence, the monodromy of f along each element of E would beconjugate to the identity, and, hence, equal to the identity, i.e. contrary to the assumption, f : X → C \ {− , } would be free homotopic to a constant.Take a component L (cid:48)(cid:48) of L ∩ X (cid:48)(cid:48) that is not contractible to the hole of X (cid:48)(cid:48) . There is an arcof ∂X (cid:48)(cid:48) between the endpoints of L (cid:48)(cid:48) such that the union ˜ L of the component L (cid:48)(cid:48) with thisarc is a closed curve in X that is not contractible and not contractible to the hole. Hence,for one of the elements of E , say for e , the intersection number of the free homotopy class (cid:98) e with the closed curve ˜ L is positive after orienting the closed curve suitably. We may takea representative γ of (cid:98) e that is contained in X (cid:48)(cid:48) . Then γ has positive intersection numberwith L (cid:48)(cid:48) . Denote the connected component of L that contains L (cid:48)(cid:48) by L . All components of L ∩ X (cid:48)(cid:48) different from L (cid:48)(cid:48) are contractible to the hole of X (cid:48)(cid:48) . Hence, γ has intersection numberzero with each of these components. Hence, γ has positive intersection number with L since˜ γ ⊂ X (cid:48)(cid:48) . We proved that the class (cid:98) e has positive intersection number with L .If (cid:98) e (cid:48) also has non-zero intersection number with L we define e (cid:48)(cid:48) = ( e (cid:48) ) ± so that theintersection number of (cid:98) e (cid:48)(cid:48) with L is positive. If (cid:98) e (cid:48) has zero intersection number with L we put e (cid:48)(cid:48) = e e (cid:48) . Then again the intersection number of (cid:98) e (cid:48)(cid:48) with L is positive. Also, theintersection number of (cid:100) e e (cid:48)(cid:48) with L is positive. The set E (cid:48) def = { e , e (cid:48)(cid:48) } satisfies the conditionrequired in the proposition. We obtained Proposition 3 for a torus with a hole.
2. A planar domain.
Let X be a planar domain. The domain X is conformally equivalentto a disc with m smoothly bounded holes, equivalently, to the Riemann sphere with m + 1smoothly bounded holes, P \ (cid:83) m +1 j =1 C j , where C m +1 contains the point ∞ . As before the basepoint of X is denoted by q , and for each j = 1 , . . . , m, the generator e j, ∈ E ⊂ π ( X, q )is represented by a curve surrounding C j once counterclockwise. We claim that there exists a component L of L with limit points on the boundary of two components ∂ C j (cid:48) and ∂ C j (cid:48)(cid:48) forsome j (cid:48) , j (cid:48)(cid:48) ∈ { , . . . , m + 1 } with j (cid:48)(cid:48) (cid:54) = j (cid:48) .Indeed, assume the contrary. Then, if a component of L has limit points on ∂ C j , j ≤ m, thenall its limit points are on ∂ C j . Take a smoothly bounded simply connected domain C (cid:48) j (cid:98) X ∪ C j that contains the closure C j , so that its boundary ∂ C (cid:48) j represents (cid:99) e j, . Then all components L (cid:48) k of L \ C (cid:48) j with an endpoint on ∂ C (cid:48) j have both endpoints on this circle. Fix a point p (cid:48) j ∈ ∂ C (cid:48) j \ L .Assign to each component L k of L \C (cid:48) j with both endpoints on ∂ C (cid:48) j the closed arc α k in ∂ C (cid:48) j \{ p (cid:48) j } with the same endpoints as L k .The arcs α k are partially ordered by inclusion. For an arc α k which contains no other ofthe arcs (a minimal arc) the curve f ◦ α k except its endpoints is contained in C \ R . Moreover,since L k is connected, the endpoints of f ◦ α k lie on f ( L k ) which is contained in one connectedcomponent of R \ {− , } . Hence, the curve f ◦ α k is homotopic in C \ {− , } (with fixedendpoints) to a curve in R \ {− , } . The function f either maps all points on ∂ C (cid:48) j \ α k thatare close to α k to the open upper half-plane or maps them all to the open lower half-plane.(Recall, that zero is a regular value of Im f .) Hence, for an open arc α (cid:48) k ⊂ ∂ C (cid:48) j \ p (cid:48) j that contains α k the curve f ◦ α (cid:48) k is homotopic in C \ {− , } (with fixed endpoints) to a curve in C \ R .Consider the arcs α k with the following property. For an open arc α (cid:48) k in ∂ C (cid:48) j \ { p (cid:48) j } whichcontains the closed arc α k the mapping f ◦ α (cid:48) k is homotopic in C \{− , } (with fixed endpoints)to a curve contained in C \ R . Induction on the arcs by inclusion shows that this property issatisfied for all maximal arcs among the α k and, hence, f | ∂ C (cid:48) j is contractible in C \ {− , } .We saw that for each hole C j , j ≤ m , whose boundary contains limit points of a connectedcomponent of L , the monodromy along the curve C (cid:48) j with base point p (cid:48) j that repesents (cid:100) e ,j istrivial. The contradiction proves the claim.With j (cid:48) and j (cid:48)(cid:48) being the numbers of the claim we consider the set E (cid:48) ⊂ E which consistsof the following primitive elements: e j (cid:48) , , the element ( e j (cid:48)(cid:48) , ) − provided j (cid:48)(cid:48) (cid:54) = m + 1, and e j (cid:48) , e j, for all j = 1 , . . . , m, j (cid:54) = j (cid:48) , j (cid:54) = j (cid:48)(cid:48) . The free homotopy class of each element of E (cid:48) has intersection number 1 with L after suitable orientation of the curve L . Each productof at most two different elements of E (cid:48) is a primitive element of π ( X, q ) and is contained in E . Moreover, the intersection number with L of the free homotopy class of any such elementequals 1 or 2. Each element of E is the product of at most two elements of E (cid:48) ∪ ( E (cid:48) ) − .The proposition is proved for the case of planar domains X . (cid:50) Proof of Proposition 4.1. A torus with a hole.
Consider the curve L and the set E (cid:48) ⊂ π ( X, q ) obtained inProposition 3. For one of the functions M l ◦ f , denoted by F , the image F ( L ) is containedin ( − , q to a point q ∈ L along a curve α in X , and consider thegenerators e = Is α ( e ) and e (cid:48) = Is α ( e (cid:48) ) of π ( X, q ), and the set Is α ( E (cid:48) ) ⊂ π ( X, q ). Then e and e (cid:48) are products of at most two elements of Is α ( E (cid:48) ). Since the free homotopy class of anelement of π ( X, q ) coincides with the free homotopy class of the element of π ( X, q ) obtainedby applying Is α , the free homotopy class of each product of one or two elements of Is α ( E (cid:48) )intersects L . We may assume as in the proof of Proposition 3 that Is α ( E (cid:48) ) consists of theelements e and e (cid:48)(cid:48) , where e (cid:48)(cid:48) is the product of at most two elements among the e and e (cid:48) andtheir inverses. Lemma 3 applies to the pair e , e (cid:48)(cid:48) , the function F , and the curve L . By theconditions of Proposition 4 and Lemma 4 of [21] the monodromies of F along e and e (cid:48)(cid:48) are notpowers of a single standard generator of the fundamental group of π ( C \ {− , } , q (cid:48) ). Since foreach ˜ e ∈ π ( X, q ) the equality A ( (cid:98) ˜ e ) = A (Is α ( (cid:98) ˜ e )) holds, this implies that the monodromyalong e and e (cid:48)(cid:48) is the product of at most two elements of L − not exceeding 2 πλ ( X ), and, hence,it has L − not exceeding 4 πλ ( X ). Since e (cid:48) is the product of at most two elements among the e and e (cid:48)(cid:48) and their inverses, we obtain Proposition 4 for e and e (cid:48) , in particular L − ( F ∗ ( e )) and L − ( F ∗ ( e (cid:48) )) do not exceed 8 πλ ( X ). Proposition 4 is proved for tori with a hole. IEMANN SURFACES OF SECOND KIND AND EFFECTIVE FINITENESS THEOREMS 19
2. A planar domain.
Consider the curve L and the set E (cid:48) of Proposition 3. Move thebase point q along an arc α to a point q ∈ L . Then f ( q ) ∈ R \ {− , } and for one of themappings f , M ◦ f , or M ◦ f , denoted by F , q (cid:48) def = F ( q ) is contained in ( − , e j = Is α ( e j, ) for each j . The e j form the basis Is α ( E ) of π ( X, q ). The set Is α ( E (cid:48) ) consistsof primitive elements of π ( X, q ) such that the free homotopy class of each product of one ortwo elements of Is α ( E (cid:48) ) intersects L . Moreover, each element of Is α ( E ) is the product of oneor two elements of Is α ( E (cid:48) ) ∪ (Is α ( E (cid:48) )) − .By the condition of the proposition not all monodromies F ∗ ( e ) , e ∈ Is α ( E (cid:48) ) , are (trivial ornon-trivial) powers of the same standard generator of π ( C \ {− , } , q (cid:48) ). Apply Lemma 3 toall pairs of elements of Is α ( E (cid:48) ) whose monodromies are not (trivial or non-trivial) powers ofthe same standard generator of π ( C \ {− , } , q (cid:48) ). Since the product of at most two differentelements of Is α ( E (cid:48) ) is contained in Is α ( E ) Lemma 3 shows that the monodromy F ∗ ( e ) alongeach element e ∈ E (cid:48) is the product of at most two factors, each with L − not exceeding 2 πλ ( X ).Since each element of Is α ( E ) is a product of at most two factors in E (cid:48) ∪ ( E (cid:48) ) − , the monodromy F ∗ ( e j ) along each generator e j of π ( X, q ) is the product of at most 4 factors of L − not exceeding2 πλ ( X ), and, hence, each monodromy F ∗ ( e j ) has L − not exceeding 8 πλ ( X ). Proposition 4is proved for planar domains.
3. The general case. Reduction to the case of non-trivial monodromies.
We firstprovide a reduction to the case when the monodromy along each element of
E ⊂ π ( X, q ) isdifferent from the identity. In other words, we will prove that provided Proposition 4 is truefor this case, it is true in the general case.Let X , f and E be as in the statement of Proposition 4. We let ˚ E be the set of elements fromthe chosen system of generators E of π ( X, q ) along which the monodromy is different fromthe identity. Keeping the same base point ˜ q of ˜ X as before we consider the quotient X ( (cid:104) ˚ E(cid:105) ) =˜ X (cid:30) (cid:104) (Is ˜ q ) − (˚ E ) (cid:105) and the covering ω (cid:104) ˚ E(cid:105) : X ( (cid:104) ˚ E(cid:105) ) → X , and the lift f (cid:104) ˚ E(cid:105) = f ◦ ω (cid:104) ˚ E(cid:105) of f to X ( (cid:104) ˚ E(cid:105) ).Put ( q ) (cid:104) ˚ E(cid:105) = ω (cid:104) ˚ E(cid:105) (˜ q ). The fundamental group of the Riemann surface X ( (cid:104) ˚ E(cid:105) ) with base point( q ) (cid:104) ˚ E(cid:105) can be identified with (cid:104) ˚ E(cid:105) . For each e j, ∈ ˚ E we will denote by ( e j, ) (cid:104) ˚ E(cid:105) the element ofthe fundamental group π ( X (cid:104) ˚ E(cid:105) ) , ( q ) (cid:104) ˚ E(cid:105) ) that corresponds to ( ω (cid:104) ˚ E(cid:105) ) ∗ (( e j, ) (cid:104) ˚ E(cid:105) ) = e j, . The lift f (cid:104) ˚ E(cid:105) to X ( (cid:104) ˚ E(cid:105) ) has non-trivial monodromy along each element of ˚ E . Our assumption impliesthat f (cid:104) ˚ E(cid:105) : X ( (cid:104) ˚ E(cid:105) ) → C \ {− , } is irreducible. Moreover, zero is a regular value of f (cid:104) ˚ E(cid:105) .Apply to the Riemann surface X ( (cid:104) ˚ E(cid:105) ) and the function f (cid:104) ˚ E(cid:105) the case of Proposition 4 when allmonodromies are non-trivial. We obtain a point q (cid:104) ˚ E(cid:105) ∈ X ( (cid:104) ˚ E(cid:105) ) and a M¨obius transformation M l , such that the function F (cid:104) ˚ E(cid:105) = M l ◦ f (cid:104) ˚ E(cid:105) maps q (cid:104) ˚ E(cid:105) to a point q (cid:48) ∈ ( − , α (cid:104) ˚ E(cid:105) in X ( (cid:104) ˚ E(cid:105) ) with initial point ( q ) (cid:104) ˚ E(cid:105) and terminating point q (cid:104) ˚ E(cid:105) , such that the following holds.For each ( e j ) (cid:104) ˚ E(cid:105) = Is α (cid:104) ˚ E(cid:105) (( e j, ) (cid:104) ˚ E(cid:105) ) ∈ Is α (cid:104) ˚ E(cid:105) (˚ E ) the monodromy ( F (cid:104) ˚ E(cid:105) ) ∗ (( e j ) (cid:104) ˚ E(cid:105) ) is the product ofat most four elements of π ( C \ {− , } , q (cid:48) ) of L − not exceeding 2 πλ ( X ( (cid:104) ˚ E(cid:105) )). In particular, L − (( F (cid:104) ˚ E(cid:105) ) ∗ (( e j ) (cid:104) ˚ E(cid:105) )) ≤ πλ ( X ( (cid:104) ˚ E(cid:105) )).Consider the projection ω (cid:104) ˚ E(cid:105) : X ( (cid:104) ˚ E(cid:105) ) → X . Put F = M l ◦ f, for the M¨obius transformationmentioned above, so that F (cid:104) ˚ E(cid:105) = F ◦ ω (cid:104) ˚ E(cid:105) . Take α = ω (cid:104) ˚ E(cid:105) ◦ α (cid:104) ˚ E(cid:105) , and q def = ω (cid:104) ˚ E(cid:105) ( q (cid:104) ˚ E(cid:105) ). For each e j, ∈ ˚ E we obtain ( F (cid:104) ˚ E(cid:105) ) ∗ (( e j, ) (cid:104) ˚ E(cid:105) ) = ( M l ◦ f ◦ ω (cid:104) ˚ E(cid:105) ) ∗ (( e j, ) (cid:104) ˚ E(cid:105) ) = F ∗ ◦ ( ω (cid:104) ˚ E(cid:105) ) ∗ (( e j, ) (cid:104) ˚ E(cid:105) ) = F ∗ ( e j, ). Further, for ( e j ) (cid:104) ˚ E(cid:105) = Is α (cid:104) ˚ E(cid:105) (( e j, ) (cid:104) ˚ E(cid:105) ) we get ( ω (cid:104) ˚ E(cid:105) ) ∗ (( e j ) (cid:104) ˚ E(cid:105) ) = Is α ( e j, ). Thiscan be checked by applying ω (cid:104) ˚ E(cid:105) to curves representing ( e j, ) (cid:104) ˚ E(cid:105) . Hence, ( F (cid:104) ˚ E(cid:105) ) ∗ (( e j ) (cid:104) ˚ E(cid:105) ) = F ∗ (Is α ( e j, )) = F ∗ ( e j ) for each e j ∈ Is α (˚ E ). We obtain the inequality L − ( F ∗ ( e j )) ≤ πλ ( X ( (cid:104) ˚ E(cid:105) )) (15) for each e j ∈ Is α (˚ E ), and, hence, for each e j ∈ Is α ( E ), since L − ( e j ) = 0 for e j ∈ Is α ( E \ ˚ E ).It remains to prove the inequality λ ( X ( (cid:104) ˚ E(cid:105) )) ≤ λ ( X ). The quantity λ ( X ) is equal to themaximum of the extremal length of the annuli ˜ X (cid:30) (Is ˜ q ) − ( (cid:104) e (cid:105) ) over primitive elements e ofthe fundamental group π ( X, q ) of X that are products of at most four elements of E ∪ ( E ) − for the chosen set of generators E of π ( X, q ). Recall that we represented the Riemann surface X as quotient ˜ X (cid:30) (Is ˜ q ) − ( (cid:104)E(cid:105) ).The Riemann surface X ( (cid:104) ˚ E(cid:105) ) is equal to the quotient ˜ X (cid:30) (Is ˜ q ) − ( (cid:104) ˚ E(cid:105) ). The chosen set˚ E corresponds to the set ˚ E (cid:104) ˚ E(cid:105) of generators of the fundamental group π ( X ( (cid:104) ˚ E(cid:105) ) , q (cid:104) ˚ E(cid:105) ). Thequantity λ ( X ( (cid:104) ˚ E(cid:105) )) is the maximum of the extremal length of ˜ X (cid:30) (Is ˜ q ) − ( (cid:104) e (cid:105) ) over el-ements e ∈ π ( X, q ), such that ( e ) (cid:104) ˚ E(cid:105) is a primitive element of the fundamental group π ( X ( (cid:104) ˚ E(cid:105) ) , q (cid:104) ˚ E(cid:105) ) of X ( (cid:104) ˚ E(cid:105) ) which is the product of at most four elements of ˚ E (cid:104) ˚ E(cid:105) ∪ (˚ E (cid:104) ˚ E(cid:105) ) − .This quantity does not exceed λ ( X ).We proved Proposition 4 for the general case provided it is proved in the case when allmonodromies are non-trivial. Positive intersection number.
We assume now that the monodromies along all elementsof the chosen set of generators E of π ( X, q ) are non-trivial and the Riemann surface is nota torus with a hole nor a planar domain, and prove the proposition for this case. In thiscase X contains a handle that corresponds to generators, denoted by e and e (cid:48) , contained in E ⊂ π ( X, q ) so that none of the two monodromies f ∗ ( e ) or f ∗ ( e (cid:48) ) is the identity. There maybe several choices of such handles. If there is a handle so that the monodromies of f alongthe two elements from E corresponding to the handle are not powers of the same element of π ( C \ {− , } , f ( q )), we take such a handle.Consider the Riemann surface X ( (cid:104) e , e (cid:48) (cid:105) ). It is a torus with a hole and admits a holomorphiccovering ω (cid:104) e ,e (cid:48) (cid:105) : X ( (cid:104) e , e (cid:48) (cid:105) ) → X of X . Put q (cid:104) e ,e (cid:48) (cid:105) = ω (cid:104) e ,e (cid:48) (cid:105) (˜ q ). The fundamental group of X ( (cid:104) e , e (cid:48) (cid:105) ) with base point q (cid:104) e ,e (cid:48) (cid:105) can be identified with (cid:104) e , e (cid:48) (cid:105) . For an element ˜ e ∈ (cid:104) e , e (cid:48) (cid:105) we denote by (˜ e ) (cid:104) e ,e (cid:48) (cid:105) the element of π ( X ( (cid:104) e , e (cid:48) (cid:105) ) , q (cid:104) e ,e (cid:48) (cid:105) ) that projects to ˜ e under ω (cid:104) e ,e (cid:48) (cid:105) .By our conditions the lift f (cid:104) e ,e (cid:48) (cid:105) of f to the Riemann surface X ( (cid:104) e , e (cid:48) (cid:105) ) is not contractibleand 0 is a regular value of its imaginary part. Let L ⊂ X be the set where f is real. Theset ω − (cid:104) e ,e (cid:48) (cid:105) ( L ) coincides with the set of points on which the lift f (cid:104) e ,e (cid:48) (cid:105) = f ◦ ω (cid:104) e ,e (cid:48) (cid:105) of themapping f is real. According to Proposition 3 for the case of a torus with a hole there is aconnected component L (cid:104) e ,e (cid:48) (cid:105) of ω − (cid:104) e ,e (cid:48) (cid:105) ( L ) such that the free homotopy class of one of theelements ( e ) (cid:104) e ,e (cid:48) (cid:105) or ( e (cid:48) ) (cid:104) e ,e (cid:48) (cid:105) , say of the class (cid:92) ( e ) (cid:104) e ,e (cid:48) (cid:105) of loops in X (cid:104) e ,e (cid:48) (cid:105) , that are freehomotopic to loops representing ( e ) (cid:104) e ,e (cid:48) (cid:105) , has positive intersection number with L (cid:104) e ,e (cid:48) (cid:105) (aftersuitably orienting L (cid:104) e ,e (cid:48) (cid:105) ).Take a point q (cid:104) e ,e (cid:48) (cid:105) ∈ L (cid:104) e ,e (cid:48) (cid:105) . The function f ◦ ω (cid:104) e ,e (cid:48) (cid:105) takes a real value at q (cid:104) e ,e (cid:48) (cid:105) . Put q = ω (cid:104) e ,e (cid:48) (cid:105) ( q (cid:104) e ,e (cid:48) (cid:105) ). Choose a point ˜ q ∈ ˜ X , for which ω (cid:104) e ,e (cid:48) (cid:105) (˜ q ) = q (cid:104) e ,e (cid:48) (cid:105) . Let ˜ α be a curvein ˜ X with initial point ˜ q and terminating point ˜ q . Then α (cid:104) e ,e (cid:48) (cid:105) def = ω (cid:104) e ,e (cid:48) (cid:105) ( ˜ α ) is a curve in X ( (cid:104) e , e (cid:48) (cid:105) ) with initial point q (cid:104) e ,e (cid:48) (cid:105) and terminating point q (cid:104) e ,e (cid:48) (cid:105) . Note that with this choice α (cid:104) e ,e (cid:48) (cid:105) and ˜ q are compatible (as a curve in X ( (cid:104) e , e (cid:48) (cid:105) ) and a point in the universal coveringof X ( (cid:104) e , e (cid:48) (cid:105) ) that projects to the terminating point of the curve).Let ˜ E ⊂ E be any set, that contains e and e (cid:48) , for which the quotient X (cid:30) (Is ˜ q ) − ( (cid:104) ˜ E(cid:105) ) = X ( (cid:104) ˜ E(cid:105) ) is a Riemann surface of genus 1 with m (cid:48) + 1 holes. (Here m (cid:48) ≥
0. We allow thecase ˜ E = π ( X, q ), if X is of genus 1 with m + 1 holes.) We consider the projection ω (cid:104) ˜ E(cid:105)(cid:104) e ,e (cid:48) (cid:105) : X ( (cid:104) e , e (cid:48) (cid:105) ) → X ( (cid:104) ˜ E(cid:105) ). Put L (cid:104) ˜ E(cid:105) def = ω (cid:104) ˜ E(cid:105)(cid:104) e ,e (cid:48) (cid:105) ( L (cid:104) e ,e (cid:48) (cid:105) ). We write also L def = L π ( X,q ) = ω π ( X,q ) (cid:104) e ,e (cid:48) (cid:105) ( L (cid:104) e ,e (cid:48) (cid:105) ). Each L (cid:104) ˜ E(cid:105) is connected. By the commutative diagram Figure 3 the equality
IEMANN SURFACES OF SECOND KIND AND EFFECTIVE FINITENESS THEOREMS 21 ω π ( X,q ) (cid:104) ˜ E(cid:105) ◦ ω (cid:104) ˜ E(cid:105)(cid:104) e ,e (cid:48) (cid:105) = ω π ( X,q ) (cid:104) e ,e (cid:48) (cid:105) holds. Hence, ω π ( X,q ) (cid:104) ˜ E(cid:105) ( L (cid:104) ˜ E ) = L ⊂ L . (16)In other words, L (cid:104) ˜ E(cid:105) ⊂ ( ω π ( X,q ) (cid:104) ˜ E(cid:105) ) − ( L ), and therefore L (cid:104) ˜ E(cid:105) has no self-intersection.We want to show that there exists an element e ∗ of ˜ E such that the class (cid:92) ( e ∗ ) (cid:104) ˜ E(cid:105) of loops on X ( (cid:104) ˜ E(cid:105) ), that are free homotopic to representatives of ( e ∗ ) (cid:104) ˜ E(cid:105) , has positive intersection numberwith L (cid:104) ˜ E(cid:105) (with a suitable orientation of L (cid:104) ˜ E(cid:105) ).The free homotopy class (cid:92) ( e ) (cid:104) ˜ E(cid:105) in X ( (cid:104) ˜ E(cid:105) ) that is related to e intersects L (cid:104) ˜ E(cid:105) . Indeed, con-sider any loop γ (cid:48)(cid:104) ˜ E(cid:105) in X ( (cid:104) ˜ E(cid:105) ) with some base point q (cid:48)(cid:104) ˜ E(cid:105) , that represents (cid:92) ( e ) (cid:104) ˜ E(cid:105) . There exists aloop γ (cid:48)(cid:104) e ,e (cid:48) (cid:105) in X ( (cid:104) e , e (cid:48) (cid:105) ) in X ( (cid:104) e , e (cid:48) (cid:105) ) which represents (cid:92) ( e ) (cid:104) e ,e (cid:48) (cid:105) such that ω (cid:104) ˜ E(cid:105)(cid:104) e ,e (cid:48) (cid:105) ( γ (cid:48)(cid:104) e ,e (cid:48) (cid:105) ) = γ (cid:48)(cid:104) ˜ E(cid:105) . Such a curve γ (cid:48)(cid:104) e ,e (cid:48) (cid:105) can be obtained as follows. There is a loop γ (cid:48)(cid:48)(cid:104) ˜ E(cid:105) in X ( (cid:104) ˜ E(cid:105) ) with basepoint q (cid:104) ˜ E(cid:105) that represents ( e ) (cid:104) ˜ E(cid:105) , and a curve α (cid:48)(cid:104) ˜ E(cid:105) in X ( (cid:104) ˜ E(cid:105) ), such that γ (cid:48)(cid:104) ˜ E(cid:105) is homotopicwith fixed endpoint to ( α (cid:48)(cid:104) ˜ E(cid:105) ) − γ (cid:48)(cid:48)(cid:104) ˜ E(cid:105) α (cid:48)(cid:104) ˜ E(cid:105) . Consider the lift ˜ γ (cid:48)(cid:48) of γ (cid:48)(cid:48)(cid:104) ˜ E(cid:105) to ˜ X with terminatingpoint ˜ q , and the lift ˜ α (cid:48) of α (cid:48)(cid:104) ˜ E(cid:105) with initial point ˜ q . The initial point of γ (cid:48)(cid:48)(cid:104) ˜ E(cid:105) equals σ (˜ q )for the covering transformation σ = (Is ˜ q ) − ( e ). The initial point of the curve σ (( ˜ α (cid:48) ) − )˜ γ (cid:48)(cid:48) ˜ α (cid:48) is obtained from its terminating point by applying the covering transformation σ . Hence, ω (cid:104) e ,e (cid:48) (cid:105) ( σ (( ˜ α (cid:48) ) − )˜ γ (cid:48)(cid:48) ˜ α (cid:48) ) is a closed curve in X ( (cid:104) e , e (cid:48) (cid:105) that represents (cid:92) ( e ) (cid:104) e ,e (cid:48) (cid:105) and projects to( α (cid:48)(cid:104) ˜ E(cid:105) ) − γ (cid:48)(cid:48)(cid:104) ˜ E(cid:105) α (cid:48)(cid:104) ˜ E(cid:105) under ω (cid:104) ˜ E(cid:105)(cid:104) e ,e (cid:48) (cid:105) . Since γ (cid:48)(cid:104) ˜ E(cid:105) is homotopic to ( α (cid:48)(cid:104) ˜ E(cid:105) ) − γ (cid:48)(cid:48)(cid:104) ˜ E(cid:105) α (cid:48)(cid:104) ˜ E(cid:105) with fixed basepoint, it also has a lift to X ( (cid:104) e , e (cid:48) (cid:105) ) which represents (cid:92) ( e ) (cid:104) e ,e (cid:48) (cid:105) .Since (cid:92) ( e ) (cid:104) e ,e (cid:48) (cid:105) intersects L (cid:104) e ,e (cid:48) (cid:105) , the loop γ (cid:104) e ,e (cid:48) (cid:105) has an intersection point p (cid:48)(cid:104) e ,e (cid:48) (cid:105) with L (cid:104) e ,e (cid:48) (cid:105) . The point p (cid:48)(cid:104) ˜ E(cid:105) = ω (cid:104) ˜ E(cid:105)(cid:104) e ,e (cid:48) (cid:105) ( p (cid:48)(cid:104) e ,e (cid:48) (cid:105) ) is contained in γ (cid:104) ˜ E(cid:105) and in L (cid:104) ˜ E(cid:105) . We proved that thefree homotopy class (cid:92) ( e ) (cid:104) ˜ E(cid:105) in X ( (cid:104) ˜ E(cid:105) ) intersects L (cid:104) ˜ E(cid:105) .There are several possibilities for the behaviour of the curve L (cid:104) ˜ E(cid:105) . Identify L (cid:104) ˜ E(cid:105) with itsimage under the embedding X ( (cid:104) ˜ E(cid:105) ) (cid:44) → X ( (cid:104) ˜ E(cid:105) ) c into the closed torus obtained by filling theholes of X ( (cid:104) ˜ E(cid:105) ), and identify the elements of the fundamental group π ( X ( (cid:104) ˜ E(cid:105) ) , ( q ) (cid:104) ˜ E(cid:105) ) withtheir images in the fundamental group of the closed torus X ( (cid:104) ˜ E(cid:105) ) c .If L (cid:104) ˜ E(cid:105) is closed, then it does not divide X ( (cid:104) ˜ E(cid:105) ) c . Indeed, if L (cid:104) ˜ E(cid:105) did divide the closed torus X ( (cid:104) ˜ E(cid:105) ) c , it would be null-homologous in the closed torus, hence, it would be homotopic to aconstant in the closed torus, since the fundamental group of the closed torus is commutative.Then for both, e and e (cid:48) , there would exist loops representing their free homotopy classes (cid:92) ( e ) c (cid:104) ˜ E(cid:105) and (cid:92) ( e (cid:48) ) c (cid:104) ˜ E(cid:105) in X ( (cid:104) ˜ E(cid:105) ) c and not intersecting L (cid:104) ˜ E(cid:105) .Since L (cid:104) ˜ E(cid:105) does not divide the closed torus X ( (cid:104) ˜ E(cid:105) ) c , the intersection number with L (cid:104) ˜ E(cid:105) ofone of the free homotopy classes, (cid:92) ( e ) c (cid:104) ˜ E(cid:105) or (cid:92) ( e (cid:48) ) c (cid:104) ˜ E(cid:105) , on X ( (cid:104) ˜ E(cid:105) ) c is non-zero, hence one of theclasses, (cid:92) ( e ) (cid:104) ˜ E(cid:105) or (cid:92) ( e (cid:48) ) (cid:104) ˜ E(cid:105) , on X ( (cid:104) ˜ E(cid:105) ) has positive intersection number with L (cid:104) ˜ E(cid:105) after orienting L (cid:104) ˜ E(cid:105) suitably.Suppose L (cid:104) ˜ E(cid:105) is a simple relatively closed arc in X ( (cid:104) ˜ E(cid:105) ) with the whole limit set on theboundary of a single hole C . Then by the same reasoning as above, L (cid:104) ˜ E(cid:105) ∪ C does not divide theclosed torus X ( (cid:104) ˜ E(cid:105) ) c and again the intersection number with L (cid:104) ˜ E(cid:105) of one of the classes, (cid:92) ( e ) (cid:104) ˜ E(cid:105) or (cid:92) ( e (cid:48) ) (cid:104) ˜ E(cid:105) , is non-zero. If L (cid:104) ˜ E(cid:105) is a relatively closed arc in X ( (cid:104) ˜ E(cid:105) ) with limit sets on the boundary of different holes.Then one of these holes, C j , has label j not exceeding m . The free homotopy class (cid:92) ( e j, ) (cid:104) ˜ E(cid:105) corresponding to the generator e j, ∈ E ⊂ π ( X, q ) whose representives surround C j , has non-vanishing intersection number with L (cid:104) ˜ E(cid:105) . After orienting L (cid:104) ˜ E(cid:105) suitably the intersection numberof (cid:92) ( e j, ) (cid:104) ˜ E(cid:105) with L (cid:104) ˜ E(cid:105) is positive.
End of Proof.
Let again the monodromies of f along all elements of E be non-trivial,and let ˜ E be a subset of E such that the quotient X ( (cid:104) ˜ E(cid:105) ) is a torus with m (cid:48) + 1 holes. Weobtained the following objects related to X ( (cid:104) e , e (cid:48) (cid:105) ), the point q (cid:104) e ,e (cid:48) (cid:105) , the curve L (cid:104) e ,e (cid:48) (cid:105) ,the point q (cid:104) e ,e (cid:48) (cid:105) ∈ L (cid:104) e ,e (cid:48) (cid:105) , and the curve α (cid:104) e ,e (cid:48) (cid:105) . Let q (cid:104) ˜ E(cid:105) , L (cid:104) ˜ E(cid:105) , the point q (cid:104) ˜ E(cid:105) in L (cid:104) ˜ E(cid:105) ,and α (cid:104) ˜ E(cid:105) be the respective images under ω (cid:104) ˜ E(cid:105)(cid:104) e ,e (cid:48) (cid:105) . Recall that f (cid:104) ˜ E(cid:105) = f ◦ ω π ( X,q ) (cid:104) ˜ E(cid:105) and, hence f (cid:104) ˜ E(cid:105) ( L (cid:104) ˜ E(cid:105) ) = f ◦ ω π ( X,q ) (cid:104) ˜ E(cid:105) ( L (cid:104) ˜ E(cid:105) ) = f ( L ) (see (16)) is contained in the connected component of R \ {− , } that contains f ( L ). Choose the M¨obius transformation M l for which the mapping F = M l ◦ f takes L to ( − , F (cid:104) ˜ E(cid:105) = M l ◦ f (cid:104) ˜ E(cid:105) takes L (cid:104) ˜ E(cid:105) to ( − , F (cid:104) ˜ E(cid:105) ( q (cid:104) ˜ E(cid:105) ) = F ( q ), denoted by q (cid:48) , is contained in ( − , α = ω π ( X,q ) (cid:104) e ,e (cid:48) (cid:105) ◦ α (cid:104) e ,e (cid:48) (cid:105) .For e j, ∈ ˜ E we put ( e j ) (cid:104) ˜ E(cid:105) = Is α (cid:104) ˜ E(cid:105) (( e j, ) (cid:104) ˜ E(cid:105) ) and e j = Is α ( e j, ). Since α (cid:104) e ,e (cid:48) (cid:105) and ˜ q arecompatible, also α (cid:104) ˜ E(cid:105) and ˜ q are compatible, and α and ˜ q are compatible, and the equalities F ∗ ( e j ) = ( F (cid:104) ˜ E(cid:105) ) ∗ (( e j ) (cid:104) ˜ E(cid:105) ) , e j ∈ ˜ E , on monodromies hold.Take now an arbitrary element ˜ e of E . We want to prove the statement of Proposition4 for the monodromy of F along ˜ e . For each element ˜ e of E we may choose a set ˜ E ⊂ E which contains ˜ e , e , and e (cid:48) such that ˜ X (cid:30) Is ˜ q ( (cid:104) ˜ E(cid:105) ) is a Riemann surface of genus 1 with m (cid:48) + 1 holes for some m (cid:48) , ≤ m (cid:48) ≤ , and the monodromies of F along the elements ofIs α ( ˜ E ) are not powers of a single a j . Indeed, if the monodromy along the three elements˜ e def = Is α (˜ e ), e def = Is α ( e ), and e (cid:48) def = Is α ( e (cid:48) ) are not powers of a single a j then we may takethe set ˜ E = { e , e (cid:48) , ˜ e } . If all three monodromies are powers of a single a j , then there existssome element ˜ e (cid:48) of Is α ( E ) \ { e, e (cid:48) , ˜ e } such that the monodromy of f along this element is nota power of a j . Then the monodromies along e and e (cid:48) are powers of a single element, hencethe same is true for each pair of elements of E corresponding to a handle, and the elements(˜ e ) and ˜ e (cid:48) cannot form a pair of elements from E that correspond to a handle. Hence, for˜ E = { e (cid:48) , e , ˜ e , ˜ e (cid:48) } the quotient ˜ X (cid:30) Is ˜ q ( ˜ (cid:104)E(cid:105) ) has genus 1.It remains to prove that for each of the chosen ˜ E each monodromy ( F (cid:104) ˜ E(cid:105) ) ∗ (( e j ) (cid:104) ˜ E(cid:105) ), e j ∈ Is α ( ˜ E ), is the product of at most four elements of π ( C \ {− , } , q (cid:48) ) of L − not exceeding2 πλ ( X ( (cid:104) ˜ E(cid:105) ) and L − (( F (cid:104) ˜ E(cid:105) ) ∗ (( e j ) (cid:104) ˜ E(cid:105) )) ≤ πλ ( X ( (cid:104) ˜ E(cid:105) )). This follows along the same lines asthe previous proofs. We proved the existence of the element e ∗ ∈ ˜ E such that the free homotopyclass (cid:92) ( e ∗ ) (cid:104) ˜ E(cid:105) in X ( (cid:104) ˜ E(cid:105) ) has positive intersection number with L (cid:104) ˜ E(cid:105) (oriented suitably). Thenfor each other element e j, ∈ ˜ E the free homotopy class in X ( (cid:104) ˜ E(cid:105) ) corresponding to one of theelements ( e j, ) ± or e ∗ e j, , denoted by e (cid:48) j, , has positive intersection number with L (cid:104) ˜ E(cid:105) . Let˜ E (cid:48) ⊂ E be the set consisting of e ∗ and any other element e j, of ˜ E replaced by e (cid:48) j, . Then˜ E (cid:48) is a subset of the set ˜ E of primitive elements that are products of one or two elements of E ∪ E − . The free homotopy class of the product of one or two elements of ( ˜ E ) (cid:48) intersects L (cid:104) ˜ E(cid:105) positively.For each element of ˜ E (cid:48) there is another element of ˜ E (cid:48) such that the monodromies of F (cid:104) ˜ E(cid:105) along the images of the two elements under Is α are not powers of the same a j . Apply Lemma3 to such pairs of elements of ˜ E (cid:48) and notice that each element of ˜ E is the product of at mosttwo elements of ˜ E (cid:48) . We see that each monodromy ( F (cid:104) ˜ E(cid:105) ) ∗ (( e j ) (cid:104) ˜ E(cid:105) ), e j ∈ Is α ( ˜ E ), is the product IEMANN SURFACES OF SECOND KIND AND EFFECTIVE FINITENESS THEOREMS 23 of at most four elements of π ( C \ {− , } , q (cid:48) ) of L − that does not exceed 2 πλ ( X ( (cid:104) ˜ E(cid:105) )). Sinceit can be seen as above that λ ( X ( (cid:104) ˜ E(cid:105) )) ≤ λ ( X ), we obtain that the monodromy along eachelement e j ∈ Is α ( ˜ E ) is the product of at most four elements of π ( C \ {− , } , q (cid:48) ) of L − notexceeding 2 πλ ( X ), and the estimate L − ( F ∗ ( e j )) ≤ πλ ( X ) holds. The proposition is proved. (cid:50) ( g , m )-bundles over Riemann surfaces We will consider bundles whose fibers are smooth surfaces or Riemann surfaces of type( g , m ). Definition 1. (Smooth oriented ( g , m ) fiber bundles.) Let X be a smooth oriented manifoldof dimension k , let X be a smooth (oriented) manifold of dimension k + 2 and P : X → X an orientation preserving smooth proper submersion such that for each point x ∈ X thefiber P − ( x ) is a closed oriented surface of genus g . Let E be a smooth submanifold of X that intersects each fiber P − ( x ) along a set E x of m distinguished points. Then the tuple F g , m = ( X , P , E , X ) is called a smooth (oriented) fiber bundle over X with fiber a smoothclosed oriented surface of genus g with m distinguished points (for short, a smooth oriented ( g , m ) -bundle). If m = 0 the set E is the empty set and we will often denote the bundle by ( X , P , X ). If m > x → E x locally defines m smooth sections. ( g , g fiber bundles. For g = 1 and m = 0 the bundle is also called an elliptic fiber bundle.In the case when the base manifold is a Riemann surface, a holomorphic ( g , m ) fiber bundleover X is defined as follows. Definition 2.
Let X be a Riemann surface, let X be a complex surface, and P a holomorphicproper submersion from X onto X , such that each fiber P − ( x ) is a closed Riemann surface ofgenus g . Suppose E is a complex one-dimensional submanifold of X that intersects each fiber P − ( x ) along a set E x of m distinguished points. Then the tuple F g , m = ( X , P , E , X ) is calleda holomorphic ( g , m ) fiber bundle over X . Notice that for m > x → E x locally defines m holomorphic sections.Two smooth oriented (holomorphic, respectively) ( g , m ) fiber bundles, F = ( X , P , E , X )and F = ( X , P , E , X ), are called smoothly isomorphic (holomorphically isomorphic, respec-tively,) if there are smooth (holomorphic, respectively) homeomorphisms Φ : X → X and φ : X → X such that for each x ∈ X the mapping Φ maps the fiber ( P ) − ( x ) onto the fiber( P ) − ( φ ( x )) and the set of distinguished points in ( P ) − ( x ) to the set of distinguished pointsin ( P ) − ( φ ( x )). Holomorphically isomorphic bundles will be considered the same holomorphicbundles.Two smooth (oriented) ( g , m ) fiber bundles over the same oriented smooth base manifold X , F = ( X , P , E , X ), and F = ( X , P , E , X ), are called (free) isotopic if for an open interval I containing [0 ,
1] there is a smooth ( g , m ) fiber bundle ( Y , P , E , X × I ) over the base X × I (called an isotopy) with the following property. For each t ∈ [0 ,
1] we put Y t = P − ( X × { t } )and E t = E ∩ P − ( X × { t } ). The bundle F is equal to (cid:0) Y , P | Y , E , X × { } (cid:1) , and thebundle F is equal to (cid:0) Y , P | Y , E , X × { } (cid:1) .Two smooth ( g , m )-bundles are smoothly isomorphic if and only if they are isotopic (see[21]).Denote by S a reference surface of genus g with a set E ⊂ S of m distinguished points.By Ehresmann’s Fibration Theorem each smooth ( g , m )-bundle F g , m = ( X , P , E , X ) with setof distinguished points E x def = E ∩ P − ( x ) in the fiber over x is locally smoothly trivial, i.e.each point in X has a neighbourhood U ⊂ X such that the restriction of the bundle to U isisomorphic to the trivial bundle (cid:0) U × S, pr , U × E, U (cid:1) with set { x } × E of distinguished points in the fiber { x } × S over x . Here pr : U × S → U is the projection onto the first factor. Theidea of the proof of Ehresmann’s Theorem is the following. Choose smooth coordinates on U by a mapping from a rectangular box to U . Consider smooth vector fields v j on U , which forma basis of the tangent space at each point of U . Take smooth vector fields V j on P − ( U ) thatare tangent to E at points of this set and are mapped to v j by the differential of P . Such vectorfields can easily be obtained locally. To obtain the globally defined vector fields V j on P − ( U )one uses partitions of unity. The required diffeomorphism ϕ U is obtained by composing theflows of these vector fields (in any fixed order).In this way a trivialization of the bundle can be obtained over any simply connected domain.Let q be a base point in X and γ j ( t ) , t ∈ [0 , , be smooth curves in X with base point q that represent the generators e j of the fundamental group π ( X, q ). For each j let ϕ tj : P − ( q ) → P − ( γ j ( t )) , t ∈ [0 , ϕ j = Id, be a smooth family of diffeomorphisms that mapthe set of distinguished points in P − ( q ) to the set of distinguishes points in P − ( γ j ( t )). Toobtain such a family we may restrict the bundle to the closed curve given by γ j and lift therestriction to a bundle over the real axis R . The family of diffeomorphisms may be obtainedby considering Ehrenpreis’ vector field for the lifted bundle and take the flow of this vectorfield. The mapping ϕ j obtained for t = 1 is an orientation preserving self-homeomorphism ofthe fiber over q that preserves the set of distinguished points. Its isotopy class depends onlyon the homotopy class of the curve and the isotopy class of the bundle. The isotopy class ofits inverse ( ϕ j ) − is called the monodromy of the bundle along e j .The following theorem holds (see e.g. [7] and [21] ). Theorem G.
Let X be a connected finite smooth oriented surface. The set of isotopy classesof smooth oriented ( g , m ) fiber bundles on X is in one-to-one correspondence to the set ofconjugacy classes of homomorphisms from the fundamental group π ( X, q ) into the modulargroup Mod( g , m ) of a Riemann surface of genus g with m distinguished points. The modular group Mod( g , m ) is the group of isotopy classes of self-homeomorphisms of areference Riemann surface of genus g that map a reference set of m distinguished points toitself.A holomorphic bundle is called locally holomorphically trivial if it is locally holomorphicallyisomorphic to the trivial bundle. All fibers of a locally holomorphically trivial bundle areconformally equivalent to each other. Let 2 g − m >
0. Restrict a locally holomorphicallytrivial ( g , m )-bundle to a smooth closed curve γ ( t ) , t ∈ [0 , ϕ t from the fiber over γ (0) onto the fiber over γ ( t ) , t ∈ [0 , ϕ is a conformal self-map of the fiber over γ (0), hence, a periodic mapping. This implies thatthe bundle is isotrivial. A smooth (holomorphic, respectively) bundle is called isotrivial, ifit has a finite covering by the trivial bundle. Finally, if all monodromy mapping classes of asmooth bundle are periodic, then the bundle is isotopic (equivalently, smoothly isomorphic) toan isotrivial bundle. Isotrivial holomorphic bundles are locally holomorphically trivial.We explain now the notion of irreducible smooth ( g , m )-bundles. It is based on Thurston’snotion of irreducible surface homeomorphisms. Let S be a connected finite smooth orientedsurface. It is either closed or homeomorphic to a surface with a finite number of punctures.We will assume from the beginning that S is either closed or punctured.A finite non-empty set of mutually disjoint Jordan curves { C , . . . , C α } on a connected closedor punctured oriented surface S is called admissible if no C i is homotopic to a point in X , orto a puncture, or to a C j with i (cid:54) = j . Thurston calls an isotopy class of homeomorphisms m of S (in other words, a mapping class on S ) reducible if there is an admissible system of curves { C , . . . , C α } on S such that some (and, hence, each) element in m maps the system to anisotopic system. In this case we say that the system { C , . . . , C α } reduces m . A mapping classwhich is not reducible is called irreducible. IEMANN SURFACES OF SECOND KIND AND EFFECTIVE FINITENESS THEOREMS 25
Let S be a closed or punctured surface with set E of distinguished points. We say that ϕ is a self-homeomorphism of S with distinguished points E , if ϕ is a self-homeomorphism of S that maps the set of distinguished points E to itself. Notice that each self-homeomorphismof the punctured surface S \ E extends to a self-homeomorphism of the surface S with setof distinguished points E . We will sometimes identify self-homeomorphisms of S \ E andself-homeomorphism of S with set E of distinguished points.For a (connected oriented closed or punctured) surface S and a finite subset E of S a finitenon-empty set of mutually disjoint Jordan curves { C , . . . , C α } in S \ E is called admissiblefor S with set of distinguished points E if it is admissible for S \ E . An admissible system ofcurves for S with set of distinguished points E is said to reduce a mapping class m on S withset of distinguished points E , if the induced mapping class on S \ E is reduced by this systemof curves.Conjugacy classes of reducible mapping classes can be decomposed in some sense into irre-ducible components, and conjugacy classes of reducible mapping classes can be recovered fromthe irreducible components up to products of commuting Dehn twists. Conjugacy classes ofirreducible mapping classes are classified and studied.A Dehn twist about a simple closed curve γ in an oriented surface S is a mapping that isisotopic to the following one. Take a tubular neighbourhood of γ and parameterize it as around annulus A = { e − ε < | z | < } so that γ corresponds to | z | = e − ε . The mapping is anorientation preserving self-homeomorphism of S which is the identity outside A and is equalto the mapping e − εs +2 πit → e − εs +2 πi ( t + s ) for e − εs +2 πit ∈ A , i.e. s ∈ (0 , ε is a smallpositive number.Thurston’s notion of reducible mapping classes takes over to families of mapping classeson a surface of type ( g , m ), and therefore to ( g , m )-bundles. Namely, an admissible system ofcurves on a (connected oriented closed or punctured) surface S with set of m distinguishedpoints E is said to reduce a family of mapping classes m j ∈ M ( S ; ∅ , E ) if it reduces each m j .Similarly, a ( g , m )-bundle with fiber S over the base point x and set of distinguished points E ⊂ S is called reducible if there is an admissible system of curves in the fiber over the basepoint that reduces all monodromy mapping classes simultaneously. Otherwise the bundle iscalled irreducible.Reducible bundles can be decomposed into irreducible bundle components and the reduciblebundle can be recovered from the irreducible bundle components up to commuting Dehn twistsin the fiber over the base point.Let X be a finite open connected Riemann surface. By a holomorphic (smooth, respectively)(0 , n )-bundle with a section over X we mean a holomorphic (smooth, respectively) (0 , n + 1)-bundle ( X , P , E , X ), such that the complex manifold (smooth manifold, respectively) E ⊂ X is the disjoint union of two complex manifolds (smooth manifolds, respectively) ˚ E and s , where˚ E ⊂ X intersects each fiber P − ( x ) along a set ˚ E x of n points, and s ⊂ X intersects eachfiber P − ( x ) along a single point s x . We will also say, that the mapping x → s x , x ∈ X , is aholomorphic (smooth, respectively) section of the (0 , n )-bundle with set of distinguished points˚ E x in the fiber over x .A special (0 , n + 1)-bundle is a bundle over X of the form ( X × P , pr , E , X ), where pr : X × P → X is the projection onto the first factor, and the smooth submanifold E of X × P is equal to the disjoint union ˚ E ∪ s ∞ where s ∞ intersects each fiber { x } × P along the point { x } × {∞} , and the set ˚ E intersects each fiber along n points. A special (0 , n + 1)-bundle is,in particular, a (0 , n )-bundle with a section.Two smooth (0 , n )-bundles with a section (in particular, two special (0 , n + 1)-bundles) arecalled isotopic if they are isotopic as (0 , n + 1)-bundles with an isotopy that joins the sectionsof the bundles. A holomorphic (smooth, respectively) (0 , n )-bundle with a section is isotopicto a holomorphic (smooth, respectively) special (0 , n + 1)-bundle over X (see [21]).Theorem 2 is a consequence of the following theorem on (0 , Theorem 3.
Over a connected Riemann surface of genus g with m + 1 holes there are up toisotopy no more than (cid:0) · · exp(36 πλ ( X )) (cid:1) g + m irreducible holomorphic (0 , -bundles witha holomorphic section. For a reducible (0 , , x ∈ X equals P withset of distinguished points {− , , f ( x ) , ∞} for a function f which depends holomorphically(smoothly, respectively) on the points x ∈ X and does not take the values − f is reducible, iffthis (0 , , F = (cid:0) X , P , s , X (cid:1) is called a double branched covering of thespecial holomorphic (0 , (cid:0) X × P , pr , E , X (cid:1) if there exists a holomorphic mapping P : X → X × P that maps each fiber P − ( x ) of the (1 , { x }× P of the (0 , x , such that the restriction P : P − ( x ) → { x } × P is a holomorphicdouble branched covering with branch locus being the set { x } × ( ˚ E x ∪ {∞} ) = E ∩ ( { x } × P )of distinguished points in the fiber { x } × P , and P maps the distinguished point s x in the fiber P − ( x ) over x to the point { x } × {∞} in { x } × P . We will also denote ( X × P , pr , E , X )by P (( X , P , s , X )), and call the bundle ( X , P , s , X ) a lift of ( X × P , pr , E , X ). Let thefiber of the (1 , x ∈ X be Y with distinguished point s , andlet the fiber of the (0 , x be P with distinguished points ˚ E ∪ {∞} for a set˚ E ⊂ C ( C ) (cid:30) S . Then the monodromy mapping class m ∈ M ( P ; ∞ , ˚ E ) of the (0 , X is the projection of the monodromy mappingclass m ∈ M ( Y ; s, ∅ ) of the (1 , ϕ ∈ m and ϕ ∈ m such that ϕ ( P ( ζ )) = P ( ϕ ( ζ )) , ζ ∈ Y . Wewill also say that m is a lift of m . The lifts of a mapping class m ∈ M ( P ; ∞ , ˚ E ) differ bythe involution of Y , that interchanges the sheets of the double branched covering. Hence, eachclass m ∈ M ( P ; ∞ , ˚ E ) has exactly two lifts. Proposition 5.
Let X be a Riemann surface of genus g with m + 1 ≥ holes with base point x and curves γ j representing a set of generators e j ∈ π ( X, x ) .(1) Each holomorphic (1 , -bundle over X is holomorphically isomorphic to the double branchedcovering of a special holomorphic (0 , -bundle over X .(2) Vice versa, for each special holomorphic (0 , -bundle over X and each collection m j oflifts of the g + m monodromy mapping classes m j of the bundle along the γ j there exists adouble branched covering by a holomorphic (1 , -bundle with collection of monodromy mappingclasses equal to the m j . Each special holomorphic (0 , -bundle has exactly g + m non-isotopicholomorphic lifts.(3) A lift of a special (0 , -bundle is reducible if and only if the special (0 , -bundle is reducible. The proof of the proposition uses the fact that a holomorphic (1 , X is holo-morphically isomorphic to a holomorphic bundle whose fiber over each point x is a quotient C(cid:30) Λ x of the complex plane by a lattice Λ x with distinguished point 0 (cid:30) Λ x . The lattices de-pend holomorphically on the point x . To represent the fibers as branched coverings dependingholomorphically on the points in X we use embeddings of punctured tori into C by suitableversions of the Weierstraß ℘ -function. For a detailed proof of Proposition 5 see [21]. Preparation of the proof of Theorem 3.
The proof will be given in terms of braids. Let C n ( C ) = { ( z , . . . , z n ) ∈ C n : z j (cid:54) = z k for j (cid:54) = k } be the n -dimensional configuration space.The symmetrized configuration space is its quotient C n ( C ) (cid:30) S n by the diagonal action of the IEMANN SURFACES OF SECOND KIND AND EFFECTIVE FINITENESS THEOREMS 27 symmetric group S n . We write points of C n ( C ) as ordered n -tuples ( z , . . . , z n ) of points in C ,and points of C n ( C ) (cid:30) S n as unordered tuples { z , . . . , z n } of points in C . We regard geomet-ric braids on n strands with base point E n as loops in the symmetrized configuration space C n ( C ) (cid:30) S n with base point E n , and braids on n strands ( n -braids for short) with base point E n ∈ C n ( C ) (cid:30) S n as homotopy classes of loops with base point E n in C n ( C ) (cid:30) S n , equivalently,as element of the fundamental group π ( C n ( C ) (cid:30) S n , E n ) of the symmetrized configurationspace with base point E n .Each smooth mapping F : X → C n ( C ) (cid:30) S n defines a smooth special (0 , n + 1)-bundle( X × P , pr , E , X ) , where E ∩ ( { x } × P ) = { x } × ( F ( z ) ∪ {∞} ). Vice versa, for eachsmooth special (0 , n + 1)-bundle ( X × P , pr , E , X ) the mapping that assigns to each point x ∈ X the set of finite distinguished points in the fiber over x defines a smooth mapping F : X → C n ( C ) (cid:30) S n . The mapping F is holomorphic iff the bundle is holomorphic. It is calledirreducible iff the bundle is irreducible. Choose a base point q ∈ X . The restriction of themapping F to each loop with base point q defines a geometric braid with base point F ( q ).The braid represented by it is called the monodromy of the mapping F along the element ofthe fundamental group represented by the loop.The monodromy mapping classes of a special (0 , n + 1)-bundle are isotopy classes of self-homeomorphisms of the fiber P over the base point q which map the set of finite distinguishedpoints E n = F ( q ) in this fiber onto itself, and fix ∞ . Two smooth mappings F and F from X to C n ( C ) (cid:30) S n , that have equal value E n ∈ C n ( C ) (cid:30) S n at the base point q , define isotopicspecial (0 , n + 1)-bundles, iff their restrictions to each curve in X with base point q definebraids that differ by an element of the center Z n of the braid group B n on n strands (in otherwords, by a power of a full twist). Indeed, the braid group on n strands modulo its center B n (cid:30) Z n is isomorphic to the group of mapping classes of P that fix ∞ and map E n to itself.Note that for the group PB of pure braids on three strands the quotient PB (cid:30) Z isisomorphic to the fundamental group of C \ {− , } . The isomorphism maps the generators σ j (cid:30) (cid:104) ∆ (cid:105) , j = 1 ,
2, of PB (cid:30) Z to the standard generators a j , j = 1 ,
2, of the fundamentalgroup of C \ {− , } . Here (cid:104) ∆ (cid:105) denotes the group generated by ∆ which is equal to the center Z .The proof of Theorem 3 will go now along the same lines as the proof of Theorem 1 withsome modifications. Lemma H, Lemmas 4 and 5, and Theorem I below are given in terms ofbraids rather than in terms of elements of B (cid:30) Z .The following lemma and the following theorem were proved in [18]. Lemma H.
Any braid b ∈ B which is not a power of ∆ can be written in a unique way inthe form σ kj b ∆ (cid:96) (17) where j = 1 or j = 2 , k (cid:54) = 0 is an integer, (cid:96) is a (not necessarily even) integer, and b isa word in σ and σ in reduced form. If b is not the identity, then the first term of b is anon-zero even power of σ if j = 1 , and b is a non-zero even power of σ if j = 2 . For an integer j (cid:54) = 0 we put q ( j ) = j if j is even, and for odd j we denote by q ( j ) the eveninteger neighbour of j that is closest to zero. In other words, q ( j ) = j if j (cid:54) = 0 is even, and foreach odd integer j , q ( j ) = j − sgn( j ), where sgn( j ) for a non-zero integral number j equals 1if j is positive, and − j is negative. For a braid in form (17) we put ϑ ( b ) def = σ q ( k ) j b . If b is a power of ∆ we put ϑ ( b ) def = Id.Let C n ( R ) (cid:30) S n be the totally real subspace of C n ( C ) (cid:30) S n . It is defined in the same wayas C n ( C ) (cid:30) S n by replacing C by R . Take a base point E n ∈ C n ( R ) (cid:30) S n . The fundamentalgroup π ( C n ( C ) (cid:30) S n , E n ) with base point is isomorphic to the relative fundamental group π ( C n ( C ) (cid:30) S n , C n ( R ) (cid:30) S n ) . The elements of the latter group are homotopy classes of arcs in C n ( C ) (cid:30) S n with endpoints in the totally real subspace C n ( R ) (cid:30) S n of the symmetrizedconfiguration space.Let b ∈ B n be a braid. Denote by b tr the element of the relative fundamental group π ( C n ( C ) (cid:30) S n , C n ( R ) (cid:30) S n ) that corresponds to b under the mentioned group isomorphism.For a rectangle R in the plane with sides parallel to the axes we let f : R → C n ( C ) (cid:30) S n bea mapping which admits a continuous extension to the closure ¯ R (denoted again by f ) whichmaps the (open) horizontal sides into C n ( R ) (cid:30) S n . We say that the mapping represents b tr iffor each maximal vertical line segment contained in R (i.e. R intersected with a vertical linein C ) the restriction of f to the closure of the line segment represents b tr .The extremal length of a 3-braid with totally real horizontal boundary values is defined as Λ tr ( b ) = inf { λ ( R ) : R a rectangle which admits a holomorphic map to C n ( C ) (cid:30) S n that represents b tr } . (see [18].) The following theorem holds (see [18]). Theorem I.
Let b ∈ B be a (not necessarily pure) braid which is not a power of ∆ , and let w be the reduced word representing the image of ϑ ( b ) in PB (cid:30) (cid:104) ∆ (cid:105) . Then Λ tr ( b ) ≥ π · L − ( w ) , except in the case when b = σ kj ∆ (cid:96) , where j = 1 or j = 2 , k (cid:54) = 0 is an integer number, and (cid:96) isan arbitrary integer. In this case Λ tr ( b ) = 0 . The set H def = {{ z , z , z } ∈ C ( C ) (cid:30) S : the three points z , z , z are contained in a real line in the complex plane } (18)is a smooth real hypersurface of C ( C ) (cid:30) S . Indeed, let { z , z , z } be a point of the sym-metrized configuration space. Introduce coordinates near this point by lifting a neighbourhoodof the point to C ( C ) with coordinates ( z , z , z ). Since the linear map M ( z ) def = z − z z − z , z ∈ C , maps the points z and z to the real axis, the three points z , z , and z lie on a real linein the complex plane iff the imaginary part of z (cid:48) def = M ( z ) = z − z z − z vanishes. The equationIm z − z z − z = 0 in local coordinates ( z , z , z ) defines a local piece of a smooth real hypersurface.For each complex affine self-mapping M of the complex plane we consider the diagonal action M (cid:0) ( z , z , z ) (cid:1) = (cid:0) M ( z ) , M ( z ) , M ( z ) (cid:1) on points ( z , z , z ) ∈ C ( C ), and the diagonal action M (cid:0) { z , z , z } (cid:1) = { M ( z ) , M ( z ) , M ( z ) } on points { z , z , z } ∈ C ( C ) (cid:30) S .The following two lemmas replace Lemma 1 in the case of (0 , Lemma 4.
Let A be an annulus, and let F : A → C ( C ) (cid:30) S be a holomorphic mapping whoseimage is not contained in H . Suppose L A is a simple relatively closed curve in A with limitpoints on both boundary circles of A , and F ( L A ) ⊂ H . Suppose for a point q A ∈ L A the value F ( L A ) is in the totally real subspace C ( R ) (cid:30) S . Let e A ∈ π ( A, q A ) be a generator of thefundamental group of A with base point q A . If the braid b def = F ∗ ( e A ) ∈ B is different from σ kj ∆ (cid:96) (cid:48) with j equal to or , and k (cid:54) = 0 and (cid:96) (cid:48) being integers, then L − ( ϑ ( b )) ≤ πλ ( A ) . (19)Notice that the braids σ kj ∆ (cid:96) for odd (cid:96) are exceptional for Theorem I, but not exceptionalfor Lemma 4. The reason is that the braid in Lemma 4 is related to a mapping of an annulus,not merely to a mapping of a rectangle. For t ∈ [0 , ∞ ) we put log + t def = (cid:40) log t t ∈ [1 , ∞ )0 t ∈ [0 , . IEMANN SURFACES OF SECOND KIND AND EFFECTIVE FINITENESS THEOREMS 29
Lemma 5.
If the braid in Lemma equals b = σ kj σ k (cid:48) j (cid:48) ∆ (cid:96) with j and j (cid:48) equal to or to , j (cid:48) (cid:54) = j , and k and k (cid:48) being non-zero integers, and (cid:96) an even integer, then log + (3[ | k | + (3[ | k (cid:48) | ≤ πλ ( A ) . (20)Here for a non-negative number x we denote by [ x ] the smallest integer not exceeding x . Proof of Lemma 4.
By the same argument as in the proof of Lemma 1 we may assume that F extends continuously to the closure A and the curve L A is a smooth (connected) curve in A whose endpoints are on different boundary components of A . The value of F at the point q A ∈ L A is equal to the unordered triple {− , q (cid:48) , } ∈ C ( R ) (cid:30) S for a number q (cid:48) ∈ R \ {− , } .We restrict the mapping F to A \ L . Let ˜ R be a lift of A \ L A to the universal covering ˜ A of A . We consider ˜ R as curvilinear rectangle with horizontal sides being the two different lifts of L A and vertical sides being the lifts of the two boundary circles cut at the endpoints of L A .Take a closed curve γ A : [0 , → A in A with base point q A ∈ L A , that intersects L A onlyat the base point and represents the element e A ∈ π ( A, q A ). Let ˜ γ A be the lift of γ A forwhich ˜ γ A ((0 , R , and let ˜ F = ( ˜ F , ˜ F , ˜ F ) : ˜ R → C ( C ) be a lift of F toa mapping from ˜ R to the configuration space C ( C ). The continuous extension of ˜ F to ˜ R isalso denoted by ˜ F . We may choose the lift so that the value of ˜ F at the copy of q A on thelower horizontal side of ˜ R equals ( − , q (cid:48) , z ∈ ˜ R we consider the complex affinemapping A z ( ζ ) = a ( z ) ζ + b ( z ) def = − ζ − ˜ F ( z )˜ F ( z ) − ˜ F ( z ) , ζ ∈ C . Denote by ˆ F ( z ) def = A z ( F ( z )) = A z ( { ˜ F ( z ) , ˜ F ( z ) , ˜ F ( z ) } ) , z ∈ ˜ R, the result of applying A z to each of the three points of F ( z ).Then the mapping ˆ F ( z ) = { ˆ F ( z ) , ˆ F ( z ) , ˆ F ( z ) } = {− , ˆ F ( z ) , } is holomorphic on ˜ R . Since F ( L A ) ⊂ H the mapping ˆ F ( z ) takes the horizontal sides of ˜ R to the totally real subspace C ( R ) (cid:30) S of the symmetrized configuration space. Moreover, ˆ F ( z ) maps the copy of q A onthe lower side of ˜ R to {− , q (cid:48) , } . Let ˜ˆ F be the lift of ˆ F to the configuration space which takesthe value ( − , q (cid:48) ,
1) at the copy of q A on the lower side of ˜ R .The restrictions of F and of ˆ F to the curve γ A represent elements of the relative fundamentalgroup π ( C ( C ) (cid:30) S , C ( R ) (cid:30) S ). The represented elements of the relative fundamental groupdiffer by a finite number of half-twists. Indeed, for each z , the lifts ˜ F ( z ) and ˜ˆ F ( z ) differ bya complex affine mapping. Hence, ˜ˆ F (˜ γ A ( t )) = b ( t ) + a ( t ) ˜ F (˜ γ A ( t )) for continuous functions a and b on [0 ,
1] with b (0) = 0, a (0) = 1, and b (1) and a (1) real valued. Then the function b : [0 , → C is homotopic with endpoints in R to the function that is identically equal tozero. The mapping a : [0 , → C \ { } is homotopic with endpoints in R to a | a | . Hence, themappings ˆ F ( γ A ( t )) and a ( t ) | a | ( t ) F ( γ A ( t )) from [0 ,
1] to C ( C ) (cid:30) S are homotopic with endpointsin C ( R ) (cid:30) S . The statements follows.Let ω ( z ) : A \ L A → R be the conformal mapping of the curvilinear rectangle onto therectangle of the form R = { z = x + iy : x ∈ (0 , , y ∈ (0 , a ) } , that maps the lower curvilinearside of A \ L to the lower side of R . (Note that the number a is uniquely defined by ˜ R .)Put ˚ F ( z ) def = e k πa ω ( z ) ˆ F ( z ). Then, for an integer number k the restrictions F | γ A and ˚ F | γ A represent the same element of π ( C ( C ) (cid:30) S , C ( R ) (cid:30) S )), namely b tr . We represented b tr bythe holomorphic map ˚ F from the rectangle ˜ R into C ( C ) (cid:30) S that maps horizontal sides into C ( R ) (cid:30) S . Hence, Λ ( b tr ) ≤ λ ( ˜ R ) = λ ( A \ L A ) ≤ λ ( A ) . (21)For b (cid:54) = σ kj ∆ (cid:96) with j equal to 1 or 2, and k (cid:54) = 0 and (cid:96) being integers, the statement ofLemma 4 follows from Theorem I in the same way as Lemma 1 follows from Theorem F. For b = σ kj ∆ (cid:96) with k = 0 the statement is trivial since then b = Id and L − ( ϑ (Id)) = 0. To obtain the statement in the remaining case b = σ kj ∆ (cid:96) (cid:48) +13 with j equal to 1 or 2, and k and (cid:96) (cid:48) being integers, we use Lemma 5. Notice that σ ∆ = ∆ σ and σ ∆ = ∆ σ . Hence, b = σ kj σ kj (cid:48) ∆ (cid:96) (cid:48) +23 with σ j (cid:54) = σ j (cid:48) . Let ω : A → A be the two-fold unbranched covering of A by an annulus A . The equality λ ( A ) = 2 λ ( A ) holds. Let q A be a point in ω − ( q A ), andlet L q A be the lift of L A to A that contains q A . Denote by γ A the loop ω − ( γ A ) with basepoint q A . Then F ◦ ω | γ A represents b and ( b ) tr . Lemma 5 applied to σ kj σ kj (cid:48) ∆ (cid:96) (cid:48) +23 givesthe estimate 2 log + (3[ | k | ]) ≤ πλ ( A ) = 2 πλ ( A ). Since ϑ ( b ) = σ | k | ] sgn ( k ) j , the inequality (19)follows. The lemma is proved. (cid:50) Proof of Lemma 5.
By [18], Lemma 1 and Proposition 6, statement 2, Λ tr ( σ kj σ k (cid:48) j (cid:48) ∆ (cid:96) ) ≥ π (log + (3[ | k | + (3[ | k (cid:48) | . (22)Since by (21) the inequality Λ tr ( σ kj σ k (cid:48) j (cid:48) ∆ (cid:96) ) ≤ λ ( A ) holds, the lemma is proved. (cid:50) We want to emphasize that periodic braids are not non-zero powers of a σ j , so the lemma istrue also for periodic braids. For each periodic braid b of the form σ σ = σ − ∆ , ( σ σ ) = σ ∆ , σ σ = σ − ∆ , ( σ σ ) = σ ∆ , and ∆ the L − ( ϑ ( b )) vanishes. However, for instancefor the conjugate σ − k ∆ σ k = σ − k σ k ∆ of ∆ we have L − ( ϑ ( σ − k ∆ σ k )) = 2 log(3 | k | ).Another example, for the conjugate σ − k σ σ σ k of σ σ we have σ − k σ σ σ k = σ − k − ∆ σ k = σ − k − σ k ∆ . and L − ( ϑ ( σ − k σ σ σ k )) equals 2 log(3 | k | ).Notice that the lemmas and Theorem I descend to statements on elements of B (cid:30) Z ratherthan on braids. For an element b of the quotient B (cid:30) Z we put ϑ ( b ) = ϑ ( b ) for any represen-tative b ∈ B of b .Lemma 6 below is an analog of Lemma 3. It follows from Lemma 4 in the same way asLemma 3 follows from Lemma 1. Lemma 6.
Let X be a connected finite open Riemann surface, and F : X → C ( C ) (cid:30) S be aholomorphic map that is transverse to the hypersurface H in C ( C ) (cid:30) S . Suppose L is a simplerelatively closed curve in X such that F ( L ) is contained in H , and for a point q ∈ L the point F ( q ) is contained in the totally real space C ( R ) (cid:30) S . Let e (1) and e (2) be primitive elementsof π ( X, q ) . Suppose that for e = e (1) , e = e (2) , and e = e (1) e (2) the free homotopy class (cid:98) e intersects L . Then either the two monodromies of F modulo the center F ∗ ( e ( j ) ) (cid:30) Z , j = 1 , , are powers of the same element σ j (cid:30) Z of B (cid:30) Z , or each of them is the product of at mosttwo elements b and b of B (cid:30) Z with L − ( ϑ ( b j )) ≤ πλ e (1) ,e (2) , j = 1 , , where λ e (1) ,e (2) def = max { λ ( A ( (cid:100) e (1) )) , λ ( A ( (cid:100) e (2) )) , λ ( A ( (cid:92) e (1) e (2) )) } . Proof.
Let e ∈ π ( X, q ) such that (cid:98) e intersects L . As in the proof of lemma 1 there existsan annulus A , a point q A ∈ A , and a holomorphic map ω A : ( A, q A ) → ( X, q ) that represents e . Moreover, the connected component of ( ω A ) − ( L ) that contains q A has limit points onboth boundary components of A . Put F A = F ◦ ω A . By the conditions of Lemma 6 F A ( L A ) = F ( L ) ⊂ H and F A ( q A ) ∈ C ( C ) (cid:30) S . Let e A be the generator of π ( A, q A ) for which ω A ( e A ) = e . The mapping F A : A → C ( C ) (cid:30) S , the point q A and the curve L A satisfy the conditions ofLemma 4. Notice that the equality ( F A ) ∗ ( e A ) = F ∗ ( e ) holds. Hence, if F ∗ ( e ) is not a power ofa σ j then inequality (19) holds for F ∗ ( e ).Suppose the two monodromies modulo center F ∗ ( e ( j ) ) (cid:30) Z , j = 1 , , are not (trivial or non-trivial) powers of the same element σ j (cid:30) Z of B (cid:30) Z . Then at most two of the elements, IEMANN SURFACES OF SECOND KIND AND EFFECTIVE FINITENESS THEOREMS 31 F ∗ ( e (1) ) (cid:30) Z , F ∗ ( e (1) ) (cid:30) Z , and F ∗ ( e (1) e (2) ) (cid:30) Z = F ∗ ( e (1) ) (cid:30) Z · F ∗ ( e (2) ) (cid:30) Z , are powers ofan element of the form σ j (cid:30) Z .If the monodromies modulo center along two elements among e (1) , e (2) , and e (1) e (2) are not(zero or non-zero) powers of a σ j (cid:30) Z then by Lemma 4 for each of these two monodromiesmodulo center inequality (19) holds, and the third monodromy modulo center is the productof two elements of B (cid:30) Z for which inequality (19) holds. If the monodromies modulo centeralong two elements among e (1) and e (2) have the form σ kj (cid:30) Z and σ k (cid:48) j (cid:48) (cid:30) Z , then the σ j andthe σ j (cid:48) are different and k and k (cid:48) are non-zero. The third monodromy modulo center hasthe form σ ± kj σ ± k (cid:48) j (cid:48) (cid:30) Z (or the order of the two factors interchanged). Lemma 5 gives theinequality log + (3 max([ | k | ]) + log + (3 max([ | k (cid:48) | ]) ≤ πλ e (1) ,e (2) . Since L − ( ϑ ( σ ± kj )) = log(3[ k ])and L − ( ϑ ( σ ± k (cid:48) j (cid:48) )) = log(3[ k (cid:48) ]), inequality (19) follows for the other two monodromies. Thelemma is proved. (cid:50) The following lemma holds.
Lemma 7.
Let X be a connected finite open Riemann surface, and F : X → C ( C ) (cid:30) S asmooth mapping. Suppose for a base point q of X each element of π ( X, q ) can be representedby a curve with base point q whose image under F avoids H . Then all monodromies of F arepowers of the same periodic braid of period . Proof.
Take any monodromy of F . It has a power that is a pure 3-braid b , and a representativeof b avoids H . Then for some integer l the first and the last strand of b ∆ l are fixed, and arepresentative of b ∆ l avoids H . Hence, b ∆ l = Id and b = ∆ − l . We saw that the monodromyof F along each element e ∈ π ( X, q ) is a periodic braid.If a representative f : [0 , → C ( C ) (cid:30) S , f (0) = f (1) , of a 3-braid b avoids H , then theassociated permutation τ ( b ) cannot be a transposition. Indeed, assume the contrary. Thenthere is a lift ˜ f of f to C ( C ) (cid:30) S , for which ( ˜ f (1) , ˜ f (1) , ˜ f (1)) = ( ˜ f (0) , ˜ f (0) , ˜ f (0)). Let L t be the line in C that contains ˜ f ( t ) and ˜ f ( t ), and is oriented so that running along L t inpositive direction we meet first ˜ f ( t ) and then ˜ f ( t ). The point f (0) is not on L . Assumewithout loss of generality, that it is on the left of L with the chosen orientation of L . Sincefor each t ∈ [0 ,
1] the three points ˜ f ( t ) , ˜ f ( t ) and ˜ f ( t ) in C are not on a real line, the point˜ f ( t ) is on the left of L t with the chosen orientation. But the unorientated lines L and L coincide, and their orientation is opposite. This implies ˜ f (1) (cid:54) = ˜ f (0), which is a contradiction.We proved that all monodromies are periodic with period 3.There is a smooth homotopy F s , s ∈ [0 , , of F , such that F = F , each F s is different from F only on a small neighbourhood of q , each F t avoids H on this neighbourhood of q , and F ( q ) is the set of vertices of an equilateral triangle with barycenter 0. Since F and F arefree homotopic, their monodromy homomorphisms are conjugate, and it is enough to provethe statement of the lemma for F .For notational convenience we will keep the notation F for the new mapping and assume that F ( q ) is the set of vertices of an equilateral triangle with barycenter 0. The monodromy F ∗ ( e )along each element e ∈ π ( X, q ) is a periodic braid of period 3. Hence, τ ( F ∗ ( e )) is a cyclicpermutation. Consider the braid g with base point F (0) that is represented by rotation by theangle π , i.e. by the geometric braid t → e i πt F (0) , t ∈ [0 , , that avoids H . There exists aninteger k such that F ∗ ( e ) g k is a pure braid that is represented by a mapping that avoids H .Hence, F ∗ ( e ) g k represents ∆ l for some integer l . We proved that for each e ∈ π ( X, q ) themonodromy F ∗ ( e ) is represented by rotation of F (0) around the origin by the angle πj forsome integer j . The Lemma is proved. (cid:50) Let as before X be a finite open connected Riemann surface. The following proposition isthe main ingredient of the proof of Theorem 3. Let as before E ⊂ π ( X, q ) be the system ofgenerators of the fundamental group with base point q ∈ X that was chosen in Section 1. Proposition 6.
Let ( X , P , X ) be an irreducible holomorphic special (0 , -bundle over a finiteopen Riemann surface X , that is not isotopic to a locally holomorphically trivial bundle. Let F ( z ) , z ∈ X, be the set of finite distinguished points in the fiber over z . Then there exists acomplex affine mapping M and a point q ∈ X such that M ◦ F ( q ) is contained in C ( R ) (cid:30) S , andfor an arc α in X with initial point q and terminating point q and each element e j ∈ Is α q ( E ) the monodromy modulo center ( M ◦ F ) ∗ ( e j ) (cid:30) Z can be written as product of at most elements b j,k , k = 1 , , , , , , of B (cid:30) Z with L − ( ϑ ( b j,k )) ≤ πλ ( X ) . (23) Proof of Proposition 6.
Since the bundle is not isotopic to a locally holomorphically trivialbundle, it is not possible that all monodromies are powers of the same periodic braid, andby Lemma 7 the set L (see (24)) is not empty. By the Holomorphic Transversality Theorem[22] the mapping F can be approximated on relatively compact subsets of X by holomorphicmappings to the symmetrized configuration space that are transverse to H . Similarly as in theproof of Proposition 4 we will therefore assume in the following (after slightly shrinking X toa deformation retract of X and approximating F ) that F is transverse to H . Consider the set L def = { z ∈ X : F ( z ) ∈ H} . (24)
1. A torus with a hole.
Let X be a torus with a hole and let E = { e , e (cid:48) } be the chosen set ofgenerators of π ( X, q ). There exists a connected component L of L which is not contractibleand not contractible to the hole. Indeed, otherwise there would be a base point q and a curve α q that joins q with q , such that for both elements of Is α q ( E ) there would be representingloops with base point q which do not meet L , and hence, by Lemma 7 the monodromies alongboth elements would be powers of a single periodic braid of period 3.Hence, as in the proof of Proposition 4 there exists a component L of L which is a simplesmooth relatively closed curve in X such that, perhaps after switching e and e (cid:48) and orienting L suitably, the free homotopy class (cid:98) e has positive intersection number with L . Moreover,for one of the elements e (cid:48)± or e e (cid:48) , denoted by e (cid:48)(cid:48) the intersection number of (cid:98) e (cid:48)(cid:48) with L ispositive.Move the base point q to a point q ∈ L along a curve α , and consider the respectivegenerators e = Is α ( e ) and e (cid:48)(cid:48) = Is α ( e (cid:48)(cid:48) ) of the fundamental group π ( X, q ) with base point q .Since F ( L ) ⊂ H there is a complex affine mapping M such that M ◦ F ( q ) ∈ C ( R ) (cid:30) S . Since F is irreducible, the monodromy maps modulo center ( M ◦ F ) ∗ ( e ) (cid:30) Z and ( M ◦ F ) ∗ ( e (cid:48)(cid:48) ) (cid:30) Z are not powers of a single standard generator σ j (cid:30) Z of B (cid:30) Z (see Lemma 4 of [21]). Hence,the second option of Lemma 6 occurs. We obtain that each of the ( M ◦ F ) ∗ ( e ) (cid:30) Z and( M ◦ F ) ∗ ( e (cid:48)(cid:48) ) (cid:30) Z is a product of at most two elements b j of B (cid:30) S with L − ( ϑ ( b j )) ≤ πλ ( X ).Hence, ( M ◦ F ) ∗ ( e ) (cid:30) Z and ( M ◦ F ) ∗ ( e (cid:48) ) (cid:30) Z are products of at most 4 elements of B (cid:30) Z with this property. The proposition is proved for tori with a hole.
2. A planar domain.
Let X be a planar domain. We represent X as the Riemann spherewith holes C j , j = 1 , . . . , m + 1 , such that C m +1 contains ∞ . Recall, that the set E of generators e j, , j = 1 , . . . , m, of the fundamental group π ( X, q ) with base point q is chosen so that e j, is represented by a loop with base point q that surrounds C j counterclockwise and no otherhole.Since the bundle is not isotopic to a locally holomorphically trivial bundle, Lemma 7 impliesthat L (see (24)) is not empty. Moreover, there is a connected component L of L of one ofthe following kinds. Either L has limit points on the boundary of two different holes (oneof them may contain ∞ ) (first kind), or a component L has limit points on a single hole C j , and C j ∪ L divides the plane C into two connected components each of which contains ahole (maybe, only the hole containing ∞ ) (second kind), or there is a compact component L that divides C into two connected components each of which contains at least two holes (oneof them may contain ∞ ). Indeed, suppose each non-compact component of L has boundary IEMANN SURFACES OF SECOND KIND AND EFFECTIVE FINITENESS THEOREMS 33 points on the boundary of a single hole and the union of the component with the hole does notseparate the remaining holes of X , and for each compact component of L one of the connectedcomponents of its complement in X contains at most one hole. Then there exists a base point q , a curve α q in X with initial point q and terminating point q , and a representative ofeach element of Is α q ( E ) ⊂ π ( X, q ) that avoids L . Lemma 7 implies that all monodromiesmodulo center are powers of a single periodic element of B (cid:30) Z which is a contradiction.If there is a component L of first kind we may choose the same set of primitive elements E (cid:48) ⊂ E ⊂ π ( X, q ) as in the proof of Proposition 3 in the planar case. The free homotopyclass of each element of E (cid:48) and of the product of two such elements intersects L . Moreover,each element of E is the product of at most two elements of E (cid:48) . Let α q be a curve in X with initial point q and terminating point q , and M a complex affine mapping, such that( M ◦ F )( q ) ∈ C ( R ) (cid:30) S . Since M ◦ F is irreducible, the monodromies modulo center of M ◦ F along the elements of Is α ( E (cid:48) ) are not (trivial or non-trivial) powers of a single element σ j (cid:30) Z .Hence, for each element of Is α ( E (cid:48) ) there exists another element of Is α ( E (cid:48) ) so that the secondoption of Lemma 6 holds for this pair of elements of Is α ( E (cid:48) ). Therefore, the monodromymodulo center of M ◦ F along each element of Is α ( E (cid:48) ) is the product of at most two elements of B (cid:30) Z of L − not exceeding 2 πλ ( X ), and the monodromy modulo center of M ◦ F along eachelement of Is α ( E ) is the product of at most 4 elements of B (cid:30) Z of L − , each not exceeding2 πλ ( X ).Suppose there is no component of first kind but a component L of the second kind. Assumefirst that all limit points of L are on the boundary of a hole C j that does not contain ∞ . Put E (cid:48) = { e j, } ∪ ≤ k ≤ m, k (cid:54) = j { e j, e k, } . Each element of E (cid:48) is a primitive element and is the productof at most three generators contained in the set E . Further, each element of E is the productof at most three elements of E (cid:48) ∪ E (cid:48) − .The free homotopy class of each element of E (cid:48) and of each product of two different elementsof E (cid:48) intersects L . Indeed, any curve that is contained in the complement of C j ∪ L haseither winding number zero around C j (as a curve in the complex plane C ), or its windingnumber around C j coincides with the winding number around each of the holes in the boundedconnected component of C j ∪ L . On the other hand the representatives of the free homotopyclass of e j, have winding number 1 around C j and winding number 0 around each otherhole that does not contain ∞ . The representatives of the free homotopy class of e j, e k, , k ≤ m, k (cid:54) = j , have winding number 2 around C j , winding number 1 around C k , and windingnumber zero around each other hole C l , l ≤ m . The argument for products of two elements of E (cid:48) is the same.Choose a point q ∈ L , a curve α in X with initial point q and terminating point q , anda complex affine mapping M such that M ◦ F ( q ) ∈ C ( R ) (cid:30) S . Lemma 6 finishes the prooffor this case in the same way as in the case when there is a component of first kind. In thepresent case each ( M ◦ F ) ∗ (˜ e ) (cid:30) Z , ˜ e ∈ Is α ( E (cid:48) ), can be written as a product of at most 2 factors b ∈ B (cid:30) Z with L − ( ϑ ( b )) ≤ πλ ( X ). Hence, each ( M ◦ F ) ∗ ( e j ) (cid:30) Z , e j = Is α ( e j, ), can bewritten as a product of at most 6 factors b ∈ B (cid:30) Z with L − ( ϑ ( b )) ≤ πλ ( X ).Assume that the limit points of L are on the boundary of the hole that contains ∞ . Let C j and C k be holes that are contained in different components of X \ L , and let e j , and e k , be the elements of E whose representatives surround C j , and C k respectively. Denote by E (cid:48) the set that consists of the elements e j , e k , , e j , e k , , and all elements e j , e k , ˜ e with˜ e running over E \ { e j , , e k , } . Each element of E (cid:48) is the product of at most 3 elements of E ,and each element of E is the product of at most 3 elements of E (cid:48) ∪ ( E (cid:48) ) − .Each element of E (cid:48) and each product of at most two different elements of E (cid:48) intersects L .Indeed, if a closed curve is contained in one of the components of X \ L then its windingnumber around each hole contained in the other component is zero. But for all mentionedelements there is a hole in each component of X \ L such that the winding number of the freehomotopy class of the element around the hole does not vanish. Lemma 6 applies with the same meaning of q , α , and M as before. Again, each ( M ◦ F ) ∗ ( e j ) (cid:30) Z , e j = Is α ( e j, ), can bewritten as a product of at most 6 factors b ∈ B (cid:30) Z with L − ( ϑ ( b )) ≤ πλ ( X ).Suppose there are no components of L of first or second kind, but there is a connectedcomponent L of L of the third kind. Let C j be a hole contained in the bounded componentof the complement of L , and let C k , k ≤ m, be a hole that is contained in the unboundedcomponent of X \ L . Let e j , and e k , be the elements of E whose representatives surround C j , and C k respectively. Consider the set E (cid:48) consisting of the following elements: e j , e k , , e j , e k , , and e j , e k , ˜ e for each ˜ e ∈ E different from e j , and e k , . Each element of E (cid:48) is the product of at most 4 elements of E and each element of E is the product of at most 3elements of E (cid:48) . The product of two different elements of E (cid:48) is contained in E (cid:48) .The free homotopy classes of each element of E (cid:48) and of each product of two different elementsof E (cid:48) intersects L . Indeed, if a loop is contained in the bounded connected component of X \ L its winding number around the holes C j j ≤ m, contained in the unbounded componentis zero. If a loop is contained in the unbounded connected component of X \ L its windingnumbers around all holes contained in the bounded connected component are equal. But thewinding number of e j , e k , and e j , e k , around the hole C j is positive and the windingnumber around the other holes that are contained in the bounded connected component of X \ L vanishes, hence the representatives of these two element cannot be contained in theunbounded component of X \ L . Since the winding number of representatives of these elementaround C k is positive the representatives cannot be contained in the bounded component of X \ L . For representatives of the elements e j , e k , ˜ e the winding number around C j equals2, the winding number around any other hole in the bounded component of X \ L is at most1, and the winding number around C k equals 1. Hence, the free homotopy classes of thementioned elements must intersect both components of X \ L , hence they intersect L .Representatives of any product of two elements of E (cid:48) have winding number around C j atleast 3, the winding number around any other hole in the bounded component of X \ L isat most 2, and the winding number around C k equals 2. Hence, the free homotopy classes ofthese elements intersect L .For a point q ∈ L , a curve α in X joining q and q , and a complex affine mapping M for which M ◦ F ( q ) ∈ C ( C ) (cid:30) S , an application of Lemma 6 proves that in this case each( M ◦ F ) ∗ ( e j ) (cid:30) Z , e j = Is α ( e j, ), can be written as a product of at most 6 factors b ∈ B (cid:30) Z with L − ( ϑ ( b )) ≤ πλ ( X ). Proposition 6 is proved in the planar case.
3. The general case.
As in the proof of Proposition 4 we may assume that for all generators e j, the monodromy along e j, is not the identity. We may also assume that for each handle themonodromies along the two elements of E corresponding to the handle are not powers of thesame periodic element. Indeed, a power of a non-trivial periodic element of B (cid:30) Z is eitherequal to this element, or to its inverse, or to the identity. Hence, it is enough to prove thestatement of the proposition for the monodromy of one of the generators corresponding to ahandle, if both monodromies are powers of the same periodic element and are non-trivial.Let now all monodromies be non-trivial, and suppose that the monodromies along any pairof elements of E corresponding to a handle are not powers of a single periodic braid. Choose apair of elements e , e (cid:48) of E corresponding to a handle. If there is such a pair with monodromiesnot being powers of a single element of B , we take such a pair.We consider the covering ω (cid:104) e ,e (cid:48) (cid:105) : X ( (cid:104) e , e (cid:48) (cid:105) ) → X . The covering manifold X ( (cid:104) e , e (cid:48) (cid:105) ) is atorus with a hole. The preimage ( ω (cid:104) e ,e (cid:48) (cid:105) ) − ( L ) is the set where the lift F (cid:104) e ,e (cid:48) (cid:105) = F ◦ ω (cid:104) e ,e (cid:48) (cid:105) of F to X ( (cid:104) e , e (cid:48) (cid:105) ) has values in H . As in the proof of Proposition 3 for the case of a toruswith a hole we obtain a connected component L (cid:104) e ,e (cid:48) (cid:105) of ( ω (cid:104) e ,e (cid:48) (cid:105) ) − ( L ) which (after suitableorientation) has positive intersection number with the free homotopy class of the lift of one ofthe generators, say of e , and with the free homotopy class of the lift of an element e (cid:48)(cid:48) whichis one of the elements e (cid:48) ± or e e (cid:48) . IEMANN SURFACES OF SECOND KIND AND EFFECTIVE FINITENESS THEOREMS 35
Take a point q (cid:104) e ,e (cid:48) (cid:105) ∈ L (cid:104) e ,e (cid:48) (cid:105) . Choose a point ˜ q ∈ ˜ X , for which ω (cid:104) e ,e (cid:48) (cid:105) (˜ q ) = q (cid:104) e ,e (cid:48) (cid:105) . Let ˜ α be a curve in ˜ X with initial point ˜ q and terminating point ˜ q . Then α (cid:104) e ,e (cid:48) (cid:105) def = ω (cid:104) e ,e (cid:48) (cid:105) ( ˜ α ) isa curve in X ( (cid:104) e , e (cid:48) (cid:105) ) with initial point q (cid:104) e ,e (cid:48) (cid:105) and terminating point q (cid:104) e ,e (cid:48) (cid:105) , and the curve α (cid:104) e ,e (cid:48) (cid:105) in X ( (cid:104) e , e (cid:48) (cid:105) ) and the point ˜ q in the universal covering ˜ X of X ( (cid:104) e , e (cid:48) (cid:105) ) are compatible.As in the proof of Proposition 4 we take any set ˜ E ⊂ E with e , e (cid:48) ∈ ˜ E such that X ( (cid:104) ˜ E(cid:105) )is a torus with m (cid:48) + 1 holes ( m (cid:48) ≥ e ∗ ∈ ˜ E such that (cid:92) (˜ e ∗ ) (cid:104) ˜ E(cid:105) haspositive intersection number with L (cid:104) ˜ E(cid:105) = ω (cid:104) ˜ E(cid:105)(cid:104) e ,e (cid:48) (cid:105) ( L (cid:104) e ,e (cid:48) (cid:105) ). Put q (cid:104) ˜ E(cid:105) = ω (cid:104) ˜ E(cid:105)(cid:104) e ,e (cid:48) (cid:105) ( q (cid:104) e ,e (cid:48) (cid:105) ) and α (cid:104) ˜ E(cid:105) = ω (cid:104) ˜ E(cid:105)(cid:104) e ,e (cid:48) (cid:105) ( α (cid:104) e ,e (cid:48) (cid:105) ). Then the curve α (cid:104) ˜ E(cid:105) in X ( (cid:104) ˜ E(cid:105) ) and the point q (cid:104) ˜ E(cid:105) in its universalcovering ˜ X are compatible.Let q = ω π ( X,q ) (cid:104) e ,e (cid:48) (cid:105) ( q (cid:104) e ,e (cid:48) (cid:105) ), α = ω π ( X,q ) (cid:104) e ,e (cid:48) (cid:105) ◦ α (cid:104) e ,e (cid:48) (cid:105) , and L = ω π ( X,q ) (cid:104) e ,e (cid:48) (cid:105) ( L (cid:104) e ,e (cid:48) (cid:105) ). Since F ◦ ω (cid:104) e ,e (cid:48) (cid:105) maps L (cid:104) e ,e (cid:48) (cid:105) into H , F maps L into H . In particular, F ( q ) ∈ H . Hence, there isa complex affine mapping M such that M ◦ F ( q ) ∈ C ( R ) (cid:30) S .For each element ˜ e ∈ E there exists a set ˜ E ⊂ E that contains e , e (cid:48) , ˜ e , such that X ( (cid:104) ˜ E(cid:105) )is a torus with m (cid:48) + 1 holes for 0 ≤ m (cid:48) ≤ M ◦ F along the elements of Is α ( ˜ E ) are not all (trivial or non-trivial) powersof a single element σ j (cid:30) Z .With the element ˜ e ∗ ∈ ˜ E mentioned above we take the set ˜ E (cid:48) which consists of the followingelements: the element ˜ e ∗ , and for all elements ˜ e (cid:48) of ˜ E \ { ˜ e ∗ } the element ˜ e (cid:48)(cid:48) which is equal to(˜ e (cid:48) ) ± or to ˜ e ∗ ˜ e (cid:48) so that the intersection number of its free homotopy class in X ( (cid:104) ˜ E(cid:105) ) with L (cid:104) ˜ E(cid:105) is positive. Lemma 6 implies the required statement along the same lines as in the proofof Proposition 4. We obtain the statement of Proposition 6 in the general case. Proposition 6is proved. (cid:50)
Proof of Theorem 3.
Let X be a connected Riemann surface of genus g with m +1 ≥ , X is isotopic to a holo-morphic special (0 , , X , that contain a holomorphic bundle. By Lemma 4 of [21]the monodromies of such a bundle are not powers of a single element of B (cid:30) Z which is conju-gate to a σ j (cid:30) Z , but they may be powers of a single periodic element of B (cid:30) Z (equivalently,the isotopy class may contain a locally holomorphically trivial holomorphic bundle).Consider an irreducible special holomorphic (0 , X which is not isotopic to alocally holomorphically trivial bundle. Let F ( x ) , x ∈ X, be the set of finite distinguished pointsin the fiber over x . By Proposition 6 there exists a complex affine mapping M and a point q ∈ X such that M ◦ F ( q ) is contained in C ( R ) (cid:30) S , and for an arc α in X with initial point q and terminating point q and each element e j ∈ Is α q ( E ) the monodromy ( M ◦ F ) ∗ ( e j ) (cid:30) Z ofthe bundle can be written as product of at most 6 elements b j,k , k = 1 , , , , , , of B (cid:30) Z with L − ( ϑ ( b j,k )) ≤ πλ ( X ) . (25)Consider an isotopy class of special (0 , π ( X, q ) → B (cid:30) Z whose image is generated by a single periodic ele-ment of B (cid:30) Z . Up to conjugacy we may assume that this element is one of the following:Id , ∆ (cid:30) Z , ( σ σ ) (cid:30) Z , ( σ σ ) − (cid:30) Z . For each of these elements b the equality L − ( ϑ ( b )) = 0holds, hence, also in this case the inequality (25) is satisfied for each monodromy.Using Lemma 1 of [19] the number of elements of b ∈ B (cid:30) Z (including the identity), forwhich L − ( ϑ ( b )) ≤ πλ ( X ), is estimated as follows. The element w def = ϑ ( b ) ∈ B (cid:30) Z can beconsidered as a reduced word in the free group generated by a = σ (cid:30) Z and a = σ (cid:30) Z .By Lemma 1 of [19] there are no more than exp(6 πλ ( X )) + 1 ≤ exp(6 πλ ( X )) reducedwords w in a and a (including the identity) satisfying the inequality L − ( w ) ≤ πλ ( X ). For a given element w ∈ B (cid:30) Z (including the identity) we describe now all elements b of B (cid:30) Z with ϑ ( b ) = w . If w (cid:54) = Id these are the following elements. If the first termof w equals a kj with k (cid:54) = 0, then the possibilities are b = w · ( ∆ (cid:96) (cid:30) Z ) with (cid:96) = 0 or 1, b = ( σ sgn kj (cid:30) Z ) · w · ( ∆ (cid:96) (cid:30) Z ) with (cid:96) = 0 or 1, or b = ( σ ± j (cid:48) (cid:30) Z ) · w · ( ∆ (cid:96) (cid:30) Z ) with (cid:96) = 0 or1 and σ j (cid:48) (cid:54) = σ j . Hence, for w (cid:54) = Id there are 8 possible choices of elements b ∈ B (cid:30) Z with ϑ ( b ) = w .If b = Id then the choices are ∆ (cid:96) (cid:30) Z and ( σ ± j ∆ (cid:96) ) (cid:30) Z for j = 1 , , and (cid:96) = 0 or (cid:96) = 1. Theseare 10 choices. Hence, there are no more than 15 exp(6 πλ ( X )) different elements b ∈ B (cid:30) Z with L − ( ϑ ( b )) ≤ πλ ( X ).Each monodromy is the product of at most six elements b j of B (cid:30) Z with L − ( ϑ ( b j )) ≤ πλ ( X ). Hence, for each monodromy there are no more than (15 exp(6 πλ ( X ))) possiblechoices. We proved that there are up to isotopy no more than (15 exp(6 πλ ( X ))) g + m ) =(3 · · exp(36 πλ ( X ))) g + m irreducible holomorphic (0 , X . Theorem 3 is proved. (cid:50) Notice that we proved a slightly stronger statement, namely, over a Riemann surface ofgenus g with m + 1 ≥ πλ ( X ))) g + m ) isotopy classesof smooth (0 , Proof of Theorem 2.
Proposition 5 and Theorem 3 imply Theorem 2 as follows. Sup-pose an isotopy class of smooth (1 , X con-tains a holomorphic bundle. By Proposition 5 the class contains a holomorphic bundle whichis the double branched covering of a holomorphic special (0 , , , (cid:0) πλ ( X ))) (cid:1) g + m ) holomorphic special (0 , X that are either ir-reducible or isotopic to the trivial bundle.By Theorem G there are no more than (cid:0) (cid:0) exp(6 πλ ( X ))) (cid:1) g + m ) conjugacy classes ofmonodromy homomorphisms that correspond to a special holomorphic (0 , X that is either irreducible or isotopic to the trivial bundle. Each monodromy homomorphismof the holomorphic double branched covering is a lift of the respective monodromy homo-morphism of the holomorphic special (0 , , X has 2 g + m generators. Using Theorem G for (1 , g + m (cid:0) πλ ( X ))) (cid:1) g + m ) = (cid:0) · · · exp(36 πλ ( X )) (cid:1) g + m isotopy classes of(1 , (cid:50) Remark.
Each admissible system of curves on a once punctured torus consists of a single curve.This fact implies that each reducible smooth (1 , g with m + 1 ≥ , g + m non-isotopic (0 , g with m + 1 ≥ , g + m is an upper bound for the number of all reducible isotopy classes of (1 , g with m + 1 ≥ IEMANN SURFACES OF SECOND KIND AND EFFECTIVE FINITENESS THEOREMS 37
Proof of Corollary 1.
We will prove that on a punctured Riemann surface there are nonon-constant reducible holomorphic mappings to the twice punctured complex plane and thatany homotopy class of mappings from a punctured Riemann surface to the twice puncturedcomplex plane contains at most one holomorphic mapping. This implies the corollary.Recall that a holomorphic mapping f from any punctured Riemann surface X to the twicepunctured complex plane extends by Picard’s Theorem to a meromorphic function f c on theclosed Riemann surface X c . Suppose now that X is a punctured Riemann surface and that themapping f : X → C \ {− , } is reducible, i.e. it is homotopic to a mapping into a punctureddisc contained in C \ {− , } . Perhaps after composing f with a M¨obius transformation wemay suppose that this puncture equals −
1. Then the meromorphic extension f c omits thevalue 1. Indeed, if f c was equal to 1 at some puncture of X , then f would map a loop on X with non-zero winding number around the puncture to a loop in C \ {− , } with non-zerowinding number around 1 , which contradicts the fact that f is homotopic to a mapping ontoa disc punctured at − C \ {− , } . Hence, f c is a meromorphic function on acompact Riemann surface that omits a value, and, hence f is constant. Hence, on a puncturedRiemann surface there are no non-constant reducible holomorphic mappings to C \ {− , } .Suppose f and f are non-constant homotopic holomorphic mappings from the puncturedRiemann surface X to the twice punctured complex plane. Then for their meromorphic exten-sions f c and f c the functions f c − f c − X c . Indeed, suppose, for instance, that f c − k > p . Then for the boundary γ of a small dics in X around p the curve ( f − ◦ γ in C \ {− , } has index k with respect to the origin. Since f − f − C \ {− , } , the curve ( f − ◦ γ is free homotopic to ( f − ◦ γ . Hence, f − k at p . By the same reasoning we may show that f c and f c have the same divisor.Hence, they differ by a non-zero multiplicative constant. Since the functions are non-constantthey must take the value −
2. By the same reasoning as above the functions are equal to − C \ {− , } are equal. (cid:50) Proof of Corollary 2.
We have to prove, that any reducible holomorphic (1 , X is holomorphically trivial, and that two isotopic (equivalently,smoothly isomorphic) holomorphically non-trivial holomorphic (1 , X are holo-morphically isomorphic. For a simple proof of the first fact the reader may consult [21]). Thesecond fact is obtained as follows.Suppose the holomorphically non-trivial holomorphic (1 , F j , j = 1 , , have conju-gate monodromy homomorphisms. By Proposition 5 each F j is holomorphically isomorphic to adouble branched covering of a special holomorphic (0 , X × P , pr , E j , X ) def = P ( F j ).The bundles P ( F j ) are isotopic, since they have conjugate monodromy homomorphisms. Thereis a finite unramified covering ˆ P : ˆ X → X of X , such the bundles P ( F j ) have isotopic lifts( ˆ X × P , pr , ˆ E j , X ) to ˆ X for which the complex curve ˆ E j is the union of four disjoint complexcurves ˆ E kl , k = 1 , , , , each intersecting each fiber { ˆ x } × P along a single point (ˆ x, ˆ g kj (ˆ x )).The mappings ˆ X (cid:51) ˆ x → ˆ g kj (ˆ x ) are holomorphic. We may assume that ˆ g j (ˆ x ) = ∞ for each ˆ x .Define for j = 1 , , a holomorphic isomorphism of the bundle ( ˆ X × P , pr , ˆ E j , X ) by { ˆ x } × P (cid:51) (ˆ x, ζ ) → (cid:16) ˆ x, − g j (ˆ x ) − ζ ˆ g j (ˆ x ) − ˆ g j (ˆ x ) (cid:17) . The image ˆ E (cid:48) j of ˆ E j under the j -th isomorphism intersects the fiber over each ˆ x ∈ ˆ X alongthe four points (ˆ x, − , (ˆ x, , (ˆ x, ∞ ) , and (ˆ x, ˚ g j (ˆ x )) for a holomorphic function ˚ g j on ˆ X thatavoids − , ∞ . The functions ˚ g j , j = 1 , , are homotopic, since the bundles are isotopic.By Corollary 1 the functions ˚ g and ˚ g coincide. Hence, the bundles ( ˆ X × P , pr , ˆ E j , X ) are holomorphically isomorphic. This means that there is a nowhere vanishing holomorphicfunction ˆ α on ˆ X , such that for each ˆ x ∈ ˆ X the equality { ˆ x } × ˆ E (ˆ x ) = { ˆ x } × ˆ α (ˆ x ) ˆ E (ˆ x ) holds.Here ˆ E j (ˆ x ) is defined by the equality ˆ E j ∩ ( { ˆ x } × P ) = { ˆ x } × ˆ E j (ˆ x ). Define also E j ( x ) bythe equality { x } × E j ( x ) = E j ∩ ( { x } × P ). For a point x ∈ X and ˆ x , ˆ x ∈ ˆ P − ( x ) theequalities ˆ E j (ˆ x ) = ˆ E j (ˆ x ) = E j ( x ) , hold. Hence, E ( x ) = ˆ α (ˆ x ) E ( x ) = ˆ α (ˆ x ) E ( x ). For aset E ⊂ C ( C ) (cid:30) S and a complex number α the equality E = αE is possible only if α = 1, or α = − E is obtained from {− , , } by multiplication with a non-zero complex number,or e ± πi and E is obtained from the set of vertices of an equilateral triangle with barycenter0 by multiplication with a non-zero complex number.For x in a small open disc on X and x → ˆ x j ( x ) , j = 1 , , being two local inverses of ˆ P the functions x → ˆ α (ˆ x j ( x )) are two analytic functions whose ratio is contained in a finiteset, hence the ratio is constant. Since the bundles F j , and, hence, also the P ( F j ), are locallyholomorphically non-trivial, the ratio of the two functions equals 1. We saw that for eachpair of points ˆ x , ˆ x ∈ ˆ X , that project to the same point x ∈ X , ˆ α (ˆ x ) = ˆ α (ˆ x ). Put α ( x ) = ˆ α (ˆ x j ) for any point ˆ x j ∈ (ˆ P ) − ( x ). We obtain E ( x ) = α ( x ) E ( x ), that means, thebundles P ( F j ) are holomorphically isomorphic. Since the bundles F j , j = 1 , , are doublebranched coverings of the P ( F j ) and have conjugate monodromy homomorphism, they areholomorphically isomorphic. (cid:50) Proof of Proposition 1.
Denote by S α a skeleton of T α,σ ⊂ T α which is the union of twocircles each of which lifts under the covering P : C → T α to a straight line segment which isparallel to an axis in the complex plane. Denote the intersection point of the two circles by q .Put (cid:102) S α = P − ( S α ) and (cid:103) T α,σ = P − ( T α,σ ). Note that (cid:103) T α,σ is the σ -neighbourhood of (cid:102) S α .Denote by e the generator of π ( T α,σ , q ), that lifts to a vertical line segment and e (cid:48) thegenerator of π ( T α,σ , q ), that lifts to a horizontal line segment. Put E = { e, e (cid:48) } . We show firstthe inequality λ ( T α,σ ) ≤ α + 1) σ . (26)For this purpose we take any primitive element e (cid:48)(cid:48) of the fundamental group π ( T α,σ , q ) whichis the product of at most three factors, each of the factors being an element of E or the inverseof an element of E . We represent the element e (cid:48)(cid:48) by a piecewise C mapping f from aninterval [0 , l ] to the skeleton S α . We may consider f as a piecewise C mapping from thecircle R(cid:30) ( x ∼ x + l ) to the skeleton, and assume that for all points t (cid:48) of the circle where f is not smooth, f ( t (cid:48) ) = q . Let t ∈ [0 , l ] be a point for which f ( t ) (cid:54) = q . Let ˜ f be apiecewise smooth mapping from [ t , t + l ] to the universal covering C of T α ⊂ T α,σ for which P ◦ ˜ f : [ t , t + l ] → S α considered as a mapping from the circle R(cid:30) ( x ∼ x + l ) to the skeletoncoincides with f . We may take f so that the equality | ˜ f (cid:48) | = 1 holds. The mapping may bechosen so that l ≤ α + 1. (Recall that α ≥ e is primitive.)Take any t (cid:48) for which f is not smooth. We may assume that f is chosen so that the directionof ˜ f (cid:48) changes by the angle ± π at each such point. Hence, there exists a neighbourhood I ( t (cid:48) )of t (cid:48) on the circle, such that the restriction ˜ f (cid:48) | I ( t (cid:48) ) covers two sides of a square of side length σ . Denote ˜ q (cid:48) the common vertex ˜ f (cid:48) ( t (cid:48) ) of these sides, and by ˜ q (cid:48)(cid:48) the vertex of the square thatis not a vertex of one of the two sides. Replace the union of the two sides of the square thatcontain ˜ q (cid:48) by a quarter-circle of radius σ with center at the vertex ˜ q (cid:48)(cid:48) , and parameterize thelatter by t → σ e ± i σ t so that the absolute value of the derivative equals 1. Notice that thequarter-circle is shorter than the union of the two sides.Proceed in this way with all such points t (cid:48) . After a reparameterization we obtain a C mapping ˜ f of the interval [0 , l ] of length l not exceeding 2 α + 1 whose image is contained inthe union of (cid:102) S α with some quarter-circles, such that | ˜ f (cid:48) | = 1. The distance of each point of IEMANN SURFACES OF SECOND KIND AND EFFECTIVE FINITENESS THEOREMS 39 the image of ˜ f to the boundary of (cid:103) T α,σ is not smaller than σ . The mapping ˜ f is piecewise ofclass C . The normalization condition | ˜ f (cid:48) | = 1 implies | ˜ f (cid:48)(cid:48) | ≤ σ .The projection f = P ◦ ˜ f can be considered as a mapping from the circle R(cid:30) ( x ∼ x + l ) oflength l not exceeding 2 α + 1 to T α,σ , that represents the free homotopy class (cid:98) e (cid:48)(cid:48) of the chosenelement of the fundamental group.Consider the mapping x + iy → ˜ F ( x + iy ) def = ˜ f ( x ) + i ˜ f (cid:48) ( x ) y ∈ C , where x + iy runs alongthe rectangle R l = { x + iy ∈ C : x ∈ [0 , l ] , | y | ≤ σ } . The image of this mapping is containedin (cid:103) T α,σ . Since 2 ∂∂z ˜ F ( x + iy ) = 2 ˜ f (cid:48) ( x ) + i ˜ f (cid:48)(cid:48) ( x ) y and 2 ∂∂ ¯ z ˜ F ( x + iy ) = i ˜ f (cid:48)(cid:48) ( x ) y the Beltramicoefficient µ ˜ F ( x + iy ) = ∂ ¯ z ˜ F ( x + iy ) ∂z ˜ F ( x + iy ) of ˜ F satisfies the inequality | µ ˜ F ( x + iy ) | ≤ . Hence, for K = − = 2 the mapping ˜ F descends to a K -quasiconformal mapping F of the annulus A l ofextremal length λ ( A l ) = l σ ≤ (2 α +1) σ that represents the free homotopy class of the element e (cid:48)(cid:48) of the fundamental group π ( X, q ). Realize A l as an annulus in the complex plane. Let ϕ be the solution of the Beltrami equation on C with Beltrami coefficient µ ˜ F on A l and zero else.Then the mapping g = F ◦ ϕ − is a holomorphic mapping of the annulus ϕ ( A l ) of extremallength not exceeding Kλ ( A l ) ≤ α +1) σ into T α,σ that represents the chosen element of thefundamental group π ( T α,σ , q ). Inequality (26) is proved.By Theorem 1 for tori with a hole there are up to homotopy no more than 3( e πλ ( T α,σ ) ) ≤ e · π α +1 σ < e · π α +1 σ non-constant irreducible holomorphic mappings from T α,σ to thetwice punctured complex plane.We give now the proof of the lower bound. Let δ = . We consider the annulus A α,δ def = { z ∈ C : | Re z | < δ } (cid:30) ( z ∼ z + αi ). The extremal length of the annulus equals α .For any natural number j we consider all elements of π ( C \ {− , } ,
0) of the form a ± a ± . . . a ± a ± (27)containing 2 j terms, each of the form a ± j . The choice of the sign in the exponent of each termis arbitrary. There are 2 j elements of this kind. By [18] there is a relatively compact domain G in the twice punctured complex plane C \ {− , } and a positive constant C such that thefollowing holds. For each j , each element of the fundamental group of the form (27), and foreach annulus of extremal length at least 2 Cj there exists a base point q in the annulus, anda holomorphic mapping from the annulus to G that maps q to 0 and represents the element.Put j = [ α Cδ ], where [ x ] is the largest integer not exceeding a positive number x . Theneach element of the form (27) with this number j can be represented by a holomorphic map˚ g from the annulus A α,δ to G . There is a constant C that depends only on G such that themapping ˚ g satisfies the inequality | ˚ g | < C . Let g be the lift of ˚ g to a mapping on the strip { z ∈ C : | Re z | < δ } . On the thinner strip {| Re z | < δ } the derivative of g satisfies theinequality | g (cid:48) | ≤ C δ .We will associate to the holomorphic mapping ˚ g on the annulus a smooth mapping g from T α,δ to G , such that the monodromy of its class along the circle { Re z = 0 } (cid:30) ( z ∼ z + iα ) with base point 0 (cid:30) ( z ∼ z + iα ) is equal to (27) and the monodromy along { Im z =0 } (cid:30) ( z ∼ z + 1) with the same base point equals the identity. This is done as follows. Let F α = [ − , ) × [ − α , α ) ⊂ C be a fundamental domain for the projection P : C → T α . Put ∆ α,δ = F α ∩ P − ( T α,δ ). Let χ : [0 , → R be a non-decreasing function of class C with χ (0) = 0 , χ (1) = 1, χ (cid:48) (0) = χ (cid:48) (1) = 0 and | χ (cid:48) ( t ) | ≤ . Define χ : [ − δ , +3 δ ] → [0 ,
1] by χ ( t ) = χ ( δ t + ) t ∈ [ − δ , − δ ]1 t ∈ [ − δ , + δ ] χ ( − δ t + ) t ∈ [ δ , δ ] . (28) Notice that χ is a C -function that vanishes at the endpoints of the interval [ − δ , +3 δ ] togetherwith its first derivative, is non-decreasing on [ − δ , − δ ], and non-increasing on [ δ , δ ]. Put g ( z ) = χ (Re z ) g ( z ) + (1 − χ (Re z )) g (0) for z in the intersection of ∆ α,δ with {| Re z | < δ } ,and g ( z ) = g (0) for z in the rest of ∆ α,δ .Put ϕ ( z ) = ∂∂ ¯ z g ( z ) on ∆ α,δ . Since ∂∂ ¯ z χ (Re z ) = 0 for | Re z | < δ and for | Re z | > δ , thefunction ϕ ( z ) vanishes on ∆ α,δ \ Q with Q def = ([ − δ , + δ ] × [ − δ , δ ]). On Q ∩ ∆ α,δ the inequality | ϕ ( z ) | ≤ | χ (cid:48) (Re z ) | | g ( z ) − g (0) | ≤ δ · C δ | z | < δ · C · δ = 32 C δ (29)holds. Notice that the functions g and ϕ extend to p − ( T α,δ ) as continuous doubly periodicfunctions. Hence, we may consider them as functions on T α,δ . Figure 4
We want to find a small positive number (cid:15) that depends on C and C , but not on α , suchthat the following holds. For σ def = εδ there exists a solution f of the equation ∂∂ ¯ z f ( z ) = ϕ ( z ) on T α,σ such that for each z the value | f ( z ) | is smaller than the Euclidean distance in C of ± G .Then g − f is a holomorphic mapping from T α,σ to C \ {− , } whose class has monodromiesequal to (27), and to the identity, respectively. There are at least 2 α Cδ − = e ε log 210 C ασ differenthomotopy classes of such mappings on T α,σ . This will prove the lower bound.To solve the ¯ ∂ -problem on T α,εδ = T α,σ , we consider an explicit kernel function which mimicsthe Weierstraß ℘ -function. The author is grateful to Bo Berndtsson who suggested to use thiskernel function.Recall that the Weierstraß ℘ -function related to the torus T α is the doubly periodic mero-morphic function ℘ α ( ζ ) = 1 ζ + (cid:88) ( n,m ) ∈ Z \ (0 , (cid:16) ζ − n − imα ) − n + imα ) (cid:17) on C . It defines a meromorphic function on T α with a double pole at the projection of theorigin and no other pole.Put ν = + αi . Since for n + imα (cid:54) = 0 the equality1( ζ − n − imα ) − ζ − n − imα − ν ) + ν ( ζ − n − imα ) = − ν ( ζ − n − imα ) ( ζ − n − imα − ν ) , IEMANN SURFACES OF SECOND KIND AND EFFECTIVE FINITENESS THEOREMS 41 holds, and the series with these terms converges uniformly on compact sets not containingpoles, the expression ℘ να ( ζ ) = 1 ζ − ζ − ν + (cid:88) ( n,m ) ∈ Z \ (0 , (cid:16) ζ − n − imα ) − ζ − n − imα − ν ) + ν ( n + imα ) (cid:17) defines a doubly periodic meromorphic function on C with only simple poles. The functiondescends to a meromorphic function on T α with two simple poles and no other pole.Recall that the support of ϕ is contained in Q . The set Q is contained in the the 2 δ -discin C (in the Euclidean metric) around the origin. If ζ is contained in the 2 δ -disc around theorigin and z ∈ ∆ α,δ , then the point ζ − z is contained in the 2 δ -neighbourhood (in C ) of ∆ α,δ .By the choice of δ the distance of any such point ζ − z to any lattice point n + iαm except0 is larger than − δ > . Further, for z ∈ ∆ α,δ and ζ in the 2 δ -disc around the origin thedistance of the point ζ − z to any point n + iαm + ν (including the point ν ) is not smaller than − δ = . Put Q (cid:15) def = Q ∩ ∆ α,εδ = ([ − δ , + δ ] × [ − (cid:15)δ , + (cid:15)δ ]) (cid:83) ([ − (cid:15)δ , + (cid:15)δ ] × [ − δ , + δ ]). Thenthe function f ( z ) = − π (cid:90) (cid:90) Q(cid:15) ϕ ( ζ ) ℘ να ( ζ − z ) dm ( ζ ) , (30)for z in ∆ α,(cid:15)δ is holomorphic outside Q ε and satisfies the equation ∂∂ ¯ z f = ϕ on Q ε . It extendscontinuously to a doubly periodic function on p − ( T α,(cid:15)δ ) and hence descends to a continuousfunction on T α,(cid:15)δ . It remains to estimate the supremum norm of the function f on ∆ α,σ = ∆ α,εδ . The following inequality holds for z ∈ ∆ α,σ | (cid:90) (cid:90) Q (cid:15) ϕ ( ζ ) ℘ να ( ζ − z ) dm ( ζ ) | = | π (cid:90) (cid:90) Q (cid:15) ϕ ( ζ ) (cid:16) ζ − z + ( ℘ να ( ζ − z ) − ζ − z ) (cid:17) dm ( ζ ) |≤ π (cid:90) (cid:90) Q (cid:15) C δ (cid:16) | ζ − z | + C (cid:17) dm ( ζ ) . (31)We used the upper bound for ϕ and the fact that for z ∈ ∆ α,σ and ζ in Q (cid:15) the expression | ℘ να ( ζ − z ) − ζ − z | is bounded by a universal constant C . The integral of the second term onthe right hand side does not exceed C δ · C · (cid:15)δ = 6 C C (cid:15)δ . The integral (cid:82)(cid:82) Q (cid:15) | ζ − z | dm ( ζ )does not exceed the sum of the two integrals I = (cid:82)(cid:82) ( − δ, δ ) × ( − (cid:15)δ,(cid:15)δ ) | ζ − z | dm ( ζ ) , and I = (cid:82)(cid:82) ( − (cid:15)δ,(cid:15)δ ) × ( − δ,δ ) | ζ − z | dm ( ζ ) . The first integral I is largest when z = 0. Hence, itdoes not exceed (cid:90) (cid:90) | ζ | < √ (cid:15)δ | ζ | dm ( ζ ) + 4 (cid:15)δ (cid:90) δ(cid:15)δ η dη ≤ √ π(cid:15)δ + 4 (cid:15)δ log 3 (cid:15) . (32)The second integral I is smaller. We obtain the estimate | f ( z ) | ≤ C C (cid:15)δπ + 3 C πδ (2 √ π(cid:15)δ + 4 (cid:15)δ log 3 (cid:15) ) . (33)Recall that we have chosen δ = . We may choose (cid:15) > C (and, hence,only on the domain G ) so that for all ε < ε the supremum norm of ϕ is less than the distanceof ± G . The proposition is proved. (cid:50) The proof of Proposition 2 follows the same lines. It leads to solving a ¯ ∂ -problem on adomain in the complex plane which is standard. We omit the proof. References [1]
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