Mixed tête-à-tête twists as monodromies associated with holomorphic function germs
MMIXED TˆETE- `A-TˆETE TWISTS AS MONODROMIESASSOCIATED WITH HOLOMORPHIC FUNCTION GERMS
PABLO PORTILLA CUADRADO AND BALDUR SIGUR¯DSSON
Abstract.
Tˆete-`a-tˆete graphs were introduced by N. A’Campo in 2010 withthe goal of modeling the monodromy of isolated plane curves. Mixed tˆete-`a-tˆete graphs provide a generalization which define mixed tˆete-`a-tˆete twists,which are pseudo-periodic automorphisms on surfaces. We characterize themixed tˆete-`a-tˆete twists as those pseudo-periodic automorphisms that have apower which is a product of right-handed Dehn twists around disjoint simpleclosed curves, including all boundary components. It follows that the class oftˆete-`a-tˆete twists coincides with that of monodromies associated with reducedfunction germs on isolated complex surface singularities.
Contents
1. Introduction 12. The mapping class group 23. Pseudo-periodic automorphisms 34. Pure tˆete-`a-tˆete graphs 65. Mixed tˆete-`a-tˆete graphs 86. Realization theorem 11References 211.
Introduction
In [A’C] N. A’Campo introduced the notion of pure tˆete-`a-tˆete graph in orderto model monodromies of plane curves. These are metric ribbon graphs withoutunivalent vertices that satisfy a special property. If one sees the ribbon graph Γ asa strong deformation retract of a surface Σ with boundary, the tˆete-`a-tˆete propertysays that starting at any point p and walking a distance of π in any direction andalways turning right at vertices, gets you to the same point. This property definesan element in the mapping class group MCG + (Σ , ∂ Σ) which is freely periodic, andis called the tˆete-`a-tˆete twist associated with Γ.
Date : November 28, 2018.
First author supported by SVP-2013-067644 Severo Ochoa FPI grant and by project byMTM2013-45710-C2-2-P, the two of them by the Spanish Ministry of Economy and Competi-tiveness MINECO; also supported by the project ERCEA 615655 NMST Consolidator Grant.The second author is supported by the ERCEA 615655 NMST Consolidator Grant, by theBasque Government through the BERC 2014-2017 program and by the Spanish Ministry of Econ-omy and Competitiveness MINECO: BCAM Severo Ochoa excellence accreditation SEV-2013-0323. a r X i v : . [ m a t h . G T ] N ov PABLO PORTILLA CUADRADO AND BALDUR SIGUR¯DSSON
In [Gra14], C. Graf proved that if one allows univalent vertices in tˆete-`a-tˆetegraphs, then one is able to model all freely periodic mapping classes of MCG + (Σ , ∂ Σ)with positive fractional Dehn twist coefficients. In [FPP] this result was improvedby showing that one does not need to enlarge the original class of metric ribbongraphs used to prove the same theorem.However, the geometric monodromy of an isolated plane curve singularity withone branch and more than one Puiseux pair is not of finite order (see [A’C73]). Forthis purpose, the definition of mixed tˆete-`a-tˆete graph was introduced in [FPP].These are metric ribbon graphs together with a filtration and a set of locally con-stant functions that model a class of pseudo-periodic automorphisms. It was provedthat this class includes the monodromy associated with an isolated singularity ona plane curve with one branch.In this work we continue the study of (relative) mixed tˆete-`a-tˆete graphs andwe improve and generalize results from [FPP]. The main result is a completecharacterization of the mapping classes that can be modeled by a mixed tˆete-`a-tˆetegraph. This is the content of Theorem 6.7 which says
Theorem A.
Let φ : Σ → Σ be an automorphism with fixes the boundary. Thenthere exists a mixed tˆete-`a-tˆete graph in Σ inducing its mapping class in MCG + (Σ , ∂ Σ) if and only if some power of φ is a composition of right handed Dehn twists arounddisjoint simple closed curves including all boundary components. A reduced holomorphic function germ on an isolated surface singularity has anassociated Milnor fibration with a monodromy which fixes the boundary. It isknown that this monodromy is pseudo-periodic and a power of it is a compositionof right-handed Dehn twists around disjoint simple closed curves, which include allthe boundary curves. It follows that the Milnor fiber associated with such a functiongerm contains a mixed tˆete-`a-tˆete graph which defines the monodromy. Conversely,a result by Neumann and Pichon [NP07] says that any such a surface automorphismis realized as the monodromy associated with a function germ. Hence
Theorem B.
Mixed tˆete-`a-tˆete twists are precisely the monodromies associatedwith reduced function germs on isolated surface singularities.
The structure of the work is the following. In Section 2 we briefly introducenotation related to the mapping class group that we use throughout the text.In Section 3 we fix notation and conventions about pseudo-periodic automor-phisms of surfaces. Not all of this section is contained in [FPP] since we treat abroader class of pseudo-periodic automorphisms in the present text. In particularwe allow amphidrome orbits of annuli. This section is important for Section 6 whichcontains the main results.In Sections 4 and 5 we recall some necessary definitions and results about pureand mixed tˆete-`a-tˆete graphs.Section 6 starts with the statement of the main Theorem 6.7. Two Lemmaswhich are used in the proof follow and the section ends with the proof of the mainresult.The work ends with an example that depicts a mixed tˆete-`a-tˆete graph.2.
The mapping class group
Let Σ be a surface with ∂ Σ (cid:54) = ∅ . We denote by MCG(Σ) the mapping classgroup given by the automorphisms of Σ up to isotopy, where the automorphisms IXED TˆETE-`A-TˆETE TWISTS AS MONODROMIES 3 of the isotopy do not neccesarily fix the boundary. Let ∂ Σ ⊂ ∂ Σ be a subsetformed by some boundary components of Σ. We will denote by MCG(Σ , ∂ Σ) themapping class group given by automorphisms of Σ that are the identity restrictedto ∂ Σ and where isotopies are along automorphisms that are the identity on theseboundary components.Let φ : Σ → Σ be a automorphism, we denote its class in MCG(Σ) by [ φ ]. If φ | ∂ Σ = id we denote its class in MCG(Σ , ∂ Σ) by [ φ ] ∂ Σ .Given two automorphisms φ and ψ of Σ that both leave invariant some subset B ⊂ ∂ Σ such that φ | B = ψ | B , we say they are isotopic relative to the action φ | B ifthere exists a family of automorphisms of Σ that isotope them as before and suchthat any automorphism of the family has the same restriction to B as φ and ψ . Wewrite [ φ ] B,φ | B = [ ψ ] B,φ | B . We denote by MCG(Σ , B, φ | B ) the set of classes [ φ ] B,φ | B with respect to this equivalence relation. We denote by MCG + (Σ , B, φ | B ) if werestrict to automorphisms preserving orientation. Consider an automorphism φ : Σ → Σ with φ | ∂ Σ = id for some subset ∂ Σ ⊂ ∂ Σ. Let D i denote a right-handed Dehn twist around a curve parallel tothe boundary component C i ⊂ ∂ Σ. Suppose that [ φ ] ∈ MCG(Σ) is of finite order,i.e. there exists a natural number n such that [ φ ] n = [id]. Then we have that[ φ n ] ∂ Σ = [ D ] t ∂ Σ · · · [ D r ] t r ∂ Σ with t i ∈ Z . We call t i /n the fractional Dehn twistcoefficient of φ at the component C i .3. Pseudo-periodic automorphisms
We recall conventions, definitions and results about pseudo-periodic automor-phisms that we will use in the present work.
Definition 3.1.
A automorphism φ : Σ → Σ is pseudo-periodic if it is isotopicto a automorphism satisfying that there exists a finite collection of disjoint simpleclosed curves C such that(i) φ ( C ) = C .(ii) φ | Σ \C is freely isotopic to a periodic automorphism.Assuming that none of the connected components of Σ \ C is either a disk or anannulus and that the set of curves is minimal, which is always possible, we name C an admissible set of curves for φ .The following theorem is a particularization on pseudo-periodic automorphismsof the more general Corollary 13.3 in [FM12] that describes a canonical form forevery automorphism of a surface. Theorem 3.2 (Almost-Canonical Form and Canonical Form) . Let Σ be a surfacewith ∂ Σ (cid:54) = ∅ . Any pseudo-periodic map of Σ is isotopic to an automorphism in almost-canonical form , that means a automorphism φ which has an admissible setof curves C = {C i } and annular neighborhoods A = {A i } with C i ⊂ A i such that(i) φ ( A ) = A .(ii) The map φ | Σ \A is periodic.When the set C is minimal we say that φ is in canonical form. Remark 3.3.
In the case we have a pseudo-periodic automorphism of Σ thatfixes pointwise some components ∂ Σ of the boundary ∂ Σ we can always find acanonical form as follows. We can find an isotopic automorphism φ relative to ∂ Σ PABLO PORTILLA CUADRADO AND BALDUR SIGUR¯DSSON that coincides with a canonical form as in the previous theorem outside a collarneighborhood U of ∂ Σ. We may assume that there exists an isotopy connectingthe automorphism and its canonical form relative to ∂ Σ. Let φ be a pseudo-periodic automorphism in some almost-canonical form.Let C , . . . , C k be a subset of curves in C that are cyclically permuted by φ , i.e. φ ( C i ) = C i +1 for i = 1 , . . . , k − φ ( C k ) = C . Suppose that we give anorientation to C , . . . , C k so that φ | C i for i = 1 , . . . , k − amphidrome if φ | C k : C k → C is orientation reversing. Notation 3.5.
Let s, c ∈ R . We denote by D s,c the automorphism of S × I givenby by ( x, t ) (cid:55)→ ( x + st + c, t ) (we are taking S = R / Z ). Observe that D s,c ◦ D s (cid:48) ,c (cid:48) = D s + s (cid:48) ,c + c (cid:48) , D − s,c = D − s, − c . In this work we always have s ∈ Q . Remark 3.6.
We can isotope D s,c to a automorphism D ps,c that is periodic on sometubular neighborhood of the core curve S × { / } of the annulus while preservingthe action on the boundary ∂ ( S × I ):(3.7) D ps,c ( x, t ) = ( x + 3 s ( t − ) + c, t ) 0 ≤ t ≤ ( x + c, t ) ≤ t ≤ ( x + 3 s ( t − ) + c, t ) ≤ t ≤ Notation 3.8.
We denote by ˜ D s the automorphism of S × I given by(3.9) ˜ D s ( x, t ) = ( − x − s ( t − ) , − t ) 0 ≤ t ≤ ( − x, − t ) ≤ t ≤ ( − x − s ( t − ) , − t ) ≤ t ≤ s ∈ Q as well. Definition 3.10.
Let
C ⊂
Σ be a simple closed curve embedded in an orientedsurface Σ. And let A be a tubular neighborhood of C . Let D : Σ → Σ be aautomorphism of the surface with D| Σ \A = id. We say that D is a negative Dehntwist around C or a right-handed Dehn twist if there exist a parametrization η : S × I → A preserving orientation such that D = η ◦ D , ◦ η − . A positive Dehn twist is defined similarly changing D , by D − , in the formulaabove. Lemma 3.11 (Linearization. Equivalent to Lemma 2.1 in [MM11]) . Let A i be anannulus and let φ : A i → A i be a automorphism that does not exchange boundarycomponents. Suppose that φ | ∂ A i is periodic. Then, after an isotopy fixing theboundary, there exists a parametrization η : S × I → A i such that φ = η ◦ D − s, − c ◦ η − for some s ∈ Q , some c ∈ R . IXED TˆETE-`A-TˆETE TWISTS AS MONODROMIES 5
Lemma 3.12 (Specialization. Equivalent to Lemma 2.3 in [MM11]) . Let A i be anannulus and let φ : A i → A i be a automorphism that exchanges boundary compo-nents. Suppose that φ | ∂ A i is periodic. Then after an isotopy fixing the boundary,there exists a parametrization η : S × I → A i such that φ = η ◦ ˜ D − s ◦ η − for some s ∈ Q . Remark 3.13.
In the case φ | ∂ A i is the identity, we have that φ = η ◦ D s, ◦ η − for some s ∈ Z , that is φ = D s for some right-handed Dehn twist as in Definition3.10. Definition 3.14 (Screw number) . Let φ be a pseudo-periodic automorphism as inTheorem 3.2. Let n be the order of φ | Σ \A . By Remark 3.13, φ n | A i equals D| s i A i fora certain s i ∈ Z .Let α be the length of the orbit in which A i lies and let ˜ α ∈ { α, α } be thesmallest number such that φ ˜ α does not exchange the boundary components of A .We define s ( A i ) := − s i n ˜ α. We call s ( A i ) the screw number of φ at A i or at C i . Remark 3.15.
Compare Definition 3.14 with [[MM11] p.4] and with Definition 2.4in [MM11]. The original definition is due to Nielsen [[Nie44]. Sect. 12] and it doesnot depend on a canonical form for φ . Since we are restricting to automorphismsthat do not exchange boundary components of the annuli A , our definition is a bitsimpler. Lemma 3.16.
Let φ be a automorphism as in Theorem 3.2 and let {A i } ⊂ A be aset of k annuli cyclically permuted by φ , i.e. φ ( A i ) = A i +1 such that φ k does notexchange boundary components. Then there exist coordinates η i : S × I → A i for the annuli in the orbit such that η − j +1 ◦ φ ◦ η j = D − s/k, − c/k where s and c are associated to A as in Lemma 3.11.Proof. See [FPP] (cid:3)
Remark 3.17.
By Remark 3.6 we can substitute D − s/k, − c/k by D p − s/k, − c/k in theprevious lemma. Lemma 3.18.
Let φ be a automorphism as in Theorem 3.2 and let {A i } ⊂ A be aset of k annuli cyclically permuted by φ , i.e. φ ( A i ) = A i +1 such that φ k exchangesboundary components. Then there exist coordinates η i : S × I → A i for the annuli in the orbit such that η − j +1 ◦ φ ◦ η j = ˜ D − s/α where s is associated to A as in Lemma 3.12. PABLO PORTILLA CUADRADO AND BALDUR SIGUR¯DSSON
Proof.
Take a parametrization of A for φ k : A → A as in Lemma 3.12, say η : S × I → A . Define recursively η j := φ ◦ η j − ◦ ˜ D s/k (see Notation 3.5). Then,we have η − j +1 ◦ φ ◦ η j = ˜ D − s/k . Since for every j we have that η j = φ j − ◦ η ◦ ˜ D s ( j − /k we have also that η − ◦ φ ◦ η α = η − ◦ φ ◦ φ k − ◦ η ◦ ˜ D s ( k − /k = ˜ D − s/k . (cid:3) Remark 3.19.
After this proof we can check that η − k ◦ φ α ◦ η k = D − s, − c to seethat the screw number s = s ( A i ) and the parameter c modulo Z of Lemma 3.16only depend on the orbit of A i .We observe also that the numbers s and c of Lemma 3.16 satisfy (cid:96) s equals s ( A i ) and (cid:96) c is only determined modulo Z and equals the usual rotation numberrot( φ α i | η ( S ×{ } ) ) ∈ (0 , Definition 3.20.
Let C be a component of ∂ Σ and let A be a compact collarneighborhood of C in Σ. Suppose that C has a metric and total length is equal to (cid:96) . Let η : S × I → A be a parametrization of A , such that η | S ×{ } : S × { } → C is an isometry.Suppose that S has the metric induced from taking S = R /(cid:96) Z with (cid:96) ∈ R > and the standard metric on R . A boundary Dehn twist of length r ∈ R > along C is a automorphism D ηr ( C ) of Σ such that:(i) it is the identity outside A (ii) the restriction of D ηr ( C ) to A in the coordinates given by η is given by( x, t ) (cid:55)→ ( x + r · t, t ) . The isotopy type of D ηr ( C ) by isotopies fixing the action on ∂ Σ does not depend onthe parametrization η . When we write just D r ( C ), it means that we are consideringa boundary Dehn twist with respect to some parametrization η . Remark 3.21.
Given a automorphism φ of a surface Σ with ∂ Σ (cid:54) = ∅ . Let C be aconnected component of ∂ Σ such that φ | C is a rotation by c ∈ [0 , A be acompact collar neighborhood of C (isomorphic to I × C ) in Σ. Let η : S × I → A be a parametrization of A , with φ ( S × { } ) = C . Up to isotopy, we can assumethat the restriction of φ to A satisfies η − ◦ φ | A ◦ η ( x, t ) = ( x + c, t ) . Pure tˆete-`a-tˆete graphs
In this section we recall some definitions and conventions from [FPP, Section 3].
A graph Γ is a 1 dimensional finite CW-complex; unless otherwise specified agraph doesn’t have univalent vertices. A ribbon graph is a graph equipped with acyclic ordering of the edges adjacent to each vertex. With a ribbon graph, one canrecover the topology of an orientable surface with boundary, we call this surfacethe thickening of Γ. A metric ribbon graph is a ribbon graph with a metric on eachof its edges.
IXED TˆETE-`A-TˆETE TWISTS AS MONODROMIES 7
A relative metric ribbon graph is a pair (Γ , A ) with A ⊂ Γ a subgraph formedby a disjoint union of circles with the property that for each connected component A i ⊂ A , there exists a boundary component on the thickening Σ of Γ such thatit retracts to A i . The relative thickening of (Γ , A ) is the thickening of Γ minusthe cylinders corresponding to the boundary components that retract to A . Inparticular, the relative thickening, also denoted by Σ contains A as boundary. Definition 4.2 (Safe walk) . Let (Γ , A ) be a metric relative ribbon graph. A safewalk for a point p in the interior of some edge is a path γ p : R ≥ → Γ with γ p (0) = p and such that:(1) The absolute value of the speed | γ (cid:48) p | measured with the metric of Γ isconstant and equal to 1. Equivalently, the safe walk is parametrized byarc length, i.e. for s small enough d ( p, γ p ( s )) = s .(2) when γ p gets to a vertex, it continues along the next edge in the givencyclic order.(3) If p is in an edge of A , the walk γ p starts running in the opposite directionto the one indicated by A seen as boundary of Σ.An (cid:96) -safe walk is the restriction of a safe walk to the interval [0 , (cid:96) ]. If a length isnot specified when referring to a safe walk, we will understand that its length is π .The notion in (2) of continuing along the next edge in the order of e ( v ) is equiv-alent to the notion of turning to the right in every vertex for paths parallel to Γ inΣ in A’Campo’s words in [A’C]. Definition 4.3 (Tˆete-`a-tˆete property) . Let (Γ , A ) be relative metric ribbon graph.We say that Γ satisfies the (cid:96) -tˆete-`a-tˆete property , or that Γ is an (cid:96) -tˆete-`a-tˆete graph if1) For any point p ∈ Γ \ ( A ∪ v (Γ)) the two different (cid:96) -safe walks starting at p , thatwe denote by γ p , ω p , satisfy γ p ( (cid:96) ) = ω p ( (cid:96) ).2) for a point p in A \ v (Γ), the end point of the unique (cid:96) -safe walk starting at p belongs to A .In this case, we say that (Γ , A ) is a relative (cid:96) -tˆete-`a-tˆete graph . If A = ∅ , we callit a pure (cid:96) -tˆete-`a-tˆete structure or graph. If (cid:96) = π we just call it pure tˆete-`a-tˆetestructure or graph. π/ − (cid:15) π/ − (cid:15) (cid:15) Figure 4.1.
An example of a relative tˆete-`a-tˆete graph. It has 2 connectedcomponents in A (the two small circles). The length of an edge in A is 2 ε andthe length of and edge from a vertex in A to a vertex depicted as a cross is π/ − ε . PABLO PORTILLA CUADRADO AND BALDUR SIGUR¯DSSON
Notation 4.4.
Let (Γ , A ) (cid:44) → (Σ , ∂ Σ) be a relative ribbon graph properly embeddedin its thickening. Let Σ Γ be the surface that results from cutting Σ along Γ, thenΣ Γ consists of as many cylinders as there are connected components in ∂ Σ \ A . Wedenote these cylinders by ˜Σ , . . . , ˜Σ r .Let g Γ : Σ Γ → Σ be the gluing map. We denote by (cid:101) Γ i the boundary componentof the cylinder Σ i that comes from the graph (that is g Γ ( (cid:101) Γ i ) ⊂ Γ) and by C i theone coming from a boundary component of Σ (that is g Γ ( C i ) ⊂ ∂ Σ). From now on,we take the convention that C i is identified with C i × { } and that ˜Γ i is identifiedwith C i × { } . We set Σ i := g Γ ( (cid:101) Σ i ) and Γ i := g Γ ( (cid:101) Γ i ). Finally we denote g Γ ( C i )also by C i since g Γ | C i is bijective.A retraction or a product structure for a component Σ i is a parametrization r i : S × I → Σ i . For each θ ∈ S , we call r i ( { θ }× I ) a retraction line. We also say that g Γ ( r i ( { θ }× I ))is a retraction line. A relative tˆete-`a-tˆete graph (Γ , A ) (cid:44) → (Σ , ∂A ) induces a mapping class [ φ Γ ] ∂ Σ \ A, id on Σ, more specifically, an element of MCG(Σ , ∂ Σ \ A ). If a product structure isspecified, then an explicit representative φ Γ is induced. For any product structure φ Γ satisfies:1) φ Γ | Γ ( p ) = γ p ( π ), that is, it induces on the graph the same action as the tˆete-`a-tˆete property.2) the mapping class [ φ Γ ] ∈ MCG(Σ) is of finite order, we also say it is periodic.3) the fractional Dehn twist coefficients t i /n (recall 2.2) along all boundary com-ponents in Σ \ A are strictly positive.Actually, in [FPP] it is proven that 2) and 3) above characterize tˆete-`a-tˆeteautomorphisms, more concretely the following is proven: Theorem 4.6.
Let φ : Σ → Σ be an automorphism of a surface with φ | ∂ Σ = id for some non-empty subset ∂ Σ ⊂ ∂ Σ . Then there exists a relative tˆete-`a-tˆetegraph (Γ , ∂ Σ \ ∂ Σ) with [ φ Γ ] ∂ Σ = [ φ ] ∂ Σ if and only if [ φ ] ∈ MCG(Σ) is of finiteorder and all the fractional Dehn twists at boundary components in ∂ Σ are strictlypositive. It corresponds to in [FPP, Theorem 5.27]. Actually it is a consequence of thatTheorem since there, the authors consider also negative fractional Dehn twists andalso negative safe walks (which turn left instead of turning right). This is done viaa sign map ι : ∂ Σ → { + , − , } .In this work we only use the original notion of A’Campo, so this map is constant+. 5. Mixed tˆete-`a-tˆete graphs
With pure, relative and general tˆete-`a-tˆete graphs we model periodic automor-phisms. Now we extend the notion of tˆete-`a-tˆete graph to be able to model somepseudo-periodic automorphisms.Let (Γ • , A • ) be a decreasing filtration on a connected relative metric ribbongraph (Γ , A ). That is(Γ , A ) = (Γ , A ) ⊃ (Γ , A ) ⊃ · · · ⊃ (Γ d , A d ) IXED TˆETE-`A-TˆETE TWISTS AS MONODROMIES 9 where ⊃ between pairs means Γ i ⊃ Γ i +1 and A i ⊃ A i +1 , and where (Γ i , A i ) is a(possibly disconnected) relative metric ribbon graph for each i = 0 , . . . , d . We saythat d is the depth of the filtration Γ • . We assume each Γ i does not have univalentvertices and is a subgraph of Γ in the usual terminology in Graph Theory. Weobserve that since each (Γ i , A i ) is a relative metric ribbon graph, we have that A i \ A i +1 is a disjoint union of connected components homeomorphic to S .For each i = 0 , . . . , d , let δ i : Γ i → R ≥ be a locally constant map (so it is a map constant on each connected component).We put the restriction that δ (Γ ) >
0. We denote the collection of all these mapsby δ • .Let p ∈ Γ, we define c p as the largest natural number such that p ∈ Γ c p . Definition 5.1 (Mixed safe walk) . Let (Γ • , A • ) be a filtered relative metric ribbongraph. Let p ∈ Γ \ A \ v (Γ). We define a mixed safe walk γ p starting at p as aconcatenation of paths defined iteratively by the following propertiesi) γ p is a safe walk of length δ ( p ) starting at p γ := p . Let p γ := γ ( δ ) beits endpoint.ii) Suppose that γ i − p is defined and let p γi be its endpoint. – If i > c p or p γi / ∈ Γ i we stop the algorithm. – If i ≤ c p and p γi ∈ Γ i then define γ ip : [0 , δ i ( p i )] → Γ i to be a safewalk of length δ i ( p γi ) starting at p γi and going in the same directionas γ i − p .iii) Repeat step ii ) until algorithm stops.Finally, define γ p := γ kp (cid:63) · · · (cid:63) γ p , that is, the mixed safe walk starting at p is theconcatenation of all the safe walks defined in the inductive process above.As in the pure case, there are two safe walks starting at each point on Γ \ ( A ∪ v (Γ)). We denote them by γ p and ω p . Definition 5.2 (Boundary mixed safe walk) . Let (Γ • , A • ) be a filtered relativemetric ribbon graph and let p ∈ A . We define a boundary mixed safe walk b p starting at p as a concatenation of a collection of paths defined iteratively by thefollowing propertiesi) b p is a boundary safe walk of length δ ( p ) starting at p := p and goingin the direction indicated by A (as in the relative tˆete-`a-tˆete case). Let p := b p ( δ ) be its endpoint.ii) Suppose that b i − p i − is defined and let p i be its endpoint. – If i > c p or p i / ∈ Γ i we stop the algorithm. – If i ≤ c ( p ) and p i ∈ Γ i then define b ip i : [0 , δ i ( p i )] → Γ i to be a safewalk of length δ i ( p i ) starting at p i and going in the same direction as b i − p i − .iii) Repeat step ii ) until algorithm stops.Finally, define b p := b kp k (cid:63) · · · (cid:63) b p , that is, the boundary mixed safe walk startingat p is the concatenation of all the safe walks defined in the inductive process. Notation 5.3.
We call the number k in Definition 5.1 (resp. Definition 5.2), the order of the mixed safe walk (resp.boundary mixed safe walk) and denote it by o ( γ p ) (resp. o ( b p )). We denote by l ( γ p ) the length of the mixed safe walk γ p which is the sum (cid:80) o ( γ p ) j =0 δ j ( p γj ) of the lengths of all the walks involved. We consider the analogousdefinition l ( b p ).As in the pure case, two mixed safe walks starting at p ∈ Γ \ v (Γ) exist. Wedenote by ω p the mixed safe walk that starts at p but in the opposite direction tothe starting direction of γ p .Observe that since the safe walk b p is completely determined by p , for a pointin A there exists only one boundary safe walk.Now we define the relative mixed tˆete-`a-tˆete property. Definition 5.4 (Relative mixed tˆete-`a-tˆete property) . Let (Γ • , A • ) be a filteredrelative metric ribbon graph and let δ • be a set of locally constant mappings δ k :Γ k → R ≥ . We say that (Γ • , A • , δ • ) satisfies the relative mixed tˆete-`a-tˆete propertyor that it is a relative mixed tˆete-`a-tˆete graph if for every p ∈ Γ − ( v (Γ) ∪ A )I) The endpoints of γ p and ω p coincide.II) c γ p ( l ( γ p )) = c p and for every p ∈ A , we have thatIII) b p ( l ( b p )) ∈ A c p As a consequence of the two previous definitions we have:
Lemma 5.5.
Let (Γ • , A • , δ • ) be a mixed relative tˆete-`a-tˆete graph, then a) o ( ω p ) = o ( γ p ) = c p b) l ( γ p ) = l ( ω p ) for every p ∈ Γ \ v (Γ) .Proof. See [FPP]. (cid:3)
Remark 5.6.
Note that for mixed tˆete-`a-tˆete graphs it is not true that p (cid:55)→ γ p ( δ ( p )) gives a continuous mapping from Γ to Γ. Remark 5.7.
Satisfying I ) and II ) of the mixed tˆete-`a-tˆete property in Defini-tion 5.4 is equivalent to satisfying:I’) For all i = 0 , . . . , d −
1, the automorphism (cid:101) φ Γ ,i = D δ i ◦ φ Γ ,i − is compatiblewith the gluing g i , that is, g i ( x ) = g i ( y ) ⇒ g i ( (cid:101) φ Γ ,i ( x )) = g i ( (cid:101) φ Γ ,i ( y )) . Below we see the diagram which shows the construction of φ Γ . IXED TˆETE-`A-TˆETE TWISTS AS MONODROMIES 11 (5.8)Σ Γ Σ Γ Σ Γ Σ Γ Σ Γ Σ Γ Σ Γ Σ Γ Σ Γ ... ... ... ... ...Σ Γ d Σ Γ d Σ Γ d Σ Σ φ Γ , − g D δ g g φ Γ , D δ g g φ Γ , D δ g g d φ Γ ,d − D δd g d φ Γ = φ Γ ,d Remark 5.9.
The pseudo-periodic automorphism φ Γ induced by a mixed tˆete-`a-tˆete graph has negative screw numbers and positive fractional Dehn twist coef-ficients as noted in [FPP]. Actually, it was also proved in [FPP] that the screwnumber associated to an orbit of annuli A ij, , . . . , A ij,k between levels i − i ofthe filtration is(5.10) − (cid:88) k δ i (Γ ij,k ) /l (˜Γ ij, ) . Realization theorem
In this section we prove Theorem 6.7 which is the main results of this paper.It characterizes the pseudo-periodic automorphisms that can be realized by mixedtˆete-`a-tˆete graphs. First we introduce some notation and conventions.Let φ : Σ → Σ be a pseudo-periodic automorphism. For the remaining of thiswork we impose the following restrictions on φ :(i) The screw numbers are all negative.(ii) It leaves at least one boundary component pointwise fixed and the frac-tional Dehn twist coefficients at these boundary components are positive.Denote by ∂ Σ ⊂ ∂ Σ the union of the boundary components pointwise fixed by φ . We assume that φ is given in some almost-canonical form as in Remark 3.3. Notation 6.1.
We define a graph G ( φ ) associated to a given almost-canonicalform:(i) It has a vertex v for each subsurface of Σ \A whose connected componentsare cyclically permuted by φ .(ii) For each set of annuli in A permuted cyclically it has an edge connectingthe vertices corresponding to the surfaces on each side of the collection ofannuli.Let N denote the set of vertices of G ( φ ). Definition 6.2.
We say that a function L : N → Z ≥ is a filtering function for G ( φ ) if it satisfies:(i) If v, v (cid:48) ∈ N are connected by an edge, then L ( v ) (cid:54) = L ( v (cid:48) ).(ii) If v ∈ N , then either v has a neighbor v (cid:48) ∈ N with L ( v ) > L ( v (cid:48) ), or L ( v ) = 0 and Σ v contains a component of ∂ Σ.Condition ii ) above implies that for L to be a filtering function, L − (0) must onlycontain vertices corresponding to subsurfaces of Σ \A that contain some componentof ∂ Σ. That same condition assures us that L − (0) is non-empty. Definition 6.3.
Define the distance function D : N → Z ≥ as follows:(i) D ( v ) = 0 for all v with Σ v ∩ ∂ Σ (cid:54) = ∅ .(ii) D ( v ) is the distance to the set D − (0), that is the number of edges of thesmallest bamboo in G ( φ ) connecting v with some vertex in D − (0). Remark 6.4.
Take some φ : Σ → Σ in canonical form and observe that thefunction D might not be a filtering function. It can happen that there are twoadjacent vertices v, v (cid:48) ∈ N with D ( v ) = D ( v (cid:48) ) or even that there is a vertex witha loop based at it (see Example 6.13). However we modify the canonical form intoan almost-canonical form for which the function D is a filtering function:Let φ : Σ → Σ be an automorphism in canonical form such that D ( v ) = D ( v (cid:48) )for some adjacent v, v (cid:48) ∈ N . Take one edge joining v and v (cid:48) , this edges correspondsto a set of annuli A , . . . , A k being permuted cyclically by φ . For each i = 1 , . . . , k ,let η i : S → A i be parametrizations as in Lemma 3.16. Let C i ⊂ A i be the corecurves of the annuli. We distinguish two cases:1) The core curves are not amphidrome. By Remark 3.6 we can isotope φ onthe annuli A i to a automorphism ˜ φ without changing the action of φ on ∂ A i so that in the annuli η i ( S × [ , ]) it is periodic. In doing so, we canredefine the canonical form to an almost-canonical form as follows.(a) for each i = 1 , . . . , k take C i out from the set C and include η i ( S ×{ } )and η i ( S × { } ).(b) for each i = 1 , . . . , k take A i out of A and include η i ( S × [0 , / η i ( S × [ , . It is clear that this new set of data defines an almost canonical form for˜ φ and that on the corresponding G ( ˜ φ ) the vertices v and v (cid:48) are no longeradjacent since a new vertex corresponding to the surface (cid:83) i η i ( S × [ , ])appears between them.2) The core curves are amphidrome. This case is completely analogous tocase 1) with the advantage that by definition of ˜ D s inNotation 3.8, it isalready periodic in the central annuli.It is clear that after performing 1) or 2) (accordingly) for all pairs of adjacentvertices v, v (cid:48) with D ( v ) = D ( v (cid:48) ) we provide φ with an almost-canonical whosedistance function D is a filtering function. Remark 6.5.
We observe that orbits of amphidrome annuli A , . . . , A i correspondto loops in G ( φ ). So we have that after performing the modification of Remark 6.4,the almost-canonical form of φ does not have any amphidrome annuli in A . How-ever, some of the surfaces of Σ \ A are now amphidrome annuli. IXED TˆETE-`A-TˆETE TWISTS AS MONODROMIES 13
Notation 6.6.
We assume φ is in the almost-canonical form induced from thecanonical form after performing the modification described in Remark 6.4. Wedenote by ˆΣ the closure of Σ \ A in Σ. Let ˆ G ( φ ) be a graph constructed as follows:(i) It has a vertex for each connected component of Σ \ C .(ii) There are as many edges joining two vertices as curves in C intersect thetwo surfaces corresponding to those vertices.We observe that the previously defined G ( φ ) is nothing but the quotient of ˆ G ( φ )by the action induced by φ on the connected components of Σ \ A .Let ˆ N be the set of vertices of ˆ G ( φ ). Since φ permutes the surfaces in ˆΣ, itinduces a permutation of the set ˆ N which we denote by σ φ . We label the set ˆ N ,as well as the connected components of ˆΣ and the connected components of A inthe following way:(i) Label the vertices that correspond to surfaces containing components of ∂ Σ by v , , v , , . . . , v β , . Let V be the union of these vertices. Notethat σ φ ( v j, ) = v j, for all j = 1 , . . . , β .(ii) Let ˆ D : ˆ N → Z ≥ be the distance function to V , that is, ˆ D ( v ) is thenumber of edges of the smallest path in ˆ G ( φ ) that joins v with V . Let V i := ˆ D − ( i ). Observe that the permutation σ φ leaves the set V i invariant.There is a labeling of V i induced by the orbits of σ φ : suppose it has β i different orbits. For each j = 1 , . . . , β i , we label the vertices in that orbitby v ij,k with k = 1 , . . . , α j so that σ φ ( v ij,k ) = v ij,k +1 and σ φ ( v ij,α j ) = v ij, .Denote by Σ ij,k the surface in ˆΣ corresponding to the vertex v ij,k . Denote by Σ i the union of the surfaces corresponding to the vertices in V i . We denote by Σ ≤ i the union of Σ , . . . , Σ i and the annuli in between them.We recall that α j is the smallest positive number such that φ α j (Σ ij,k ) = Σ ij,k . Theorem 6.7.
Let φ : Σ → Σ be a pseudo-periodic automorphism satisfyingassumptions (i) and (ii). Then there exists a relative mixed tˆete-`a-tˆete graph (Γ • , A • , δ • ) with Γ embedded in Σ such that:(i) δ i is a constant function for each i = 1 , . . . , d .(ii) [ φ ] ∂ Σ = [ φ Γ ] ∂ Σ .(iii) φ | ∂ Σ \ ∂ Σ = φ Γ | ∂ Σ \ ∂ Σ .(iv) Filtration indexes are induced by the distance function D for the almost-canonical form induced from the canonical form by Remark 6.4. Now we state and prove Lemma 6.8 and Lemma 6.9 which are used in the proofof Theorem 6.7.
Lemma 6.8.
Let φ : Σ → Σ be a periodic automorphism of order n . Let C = C (cid:116) · · · (cid:116) C k be a non-empty collection of boundary components of Σ such that φ ( C i ) = C i , that is, each one is invariant by φ . For each i let m i be a metric on C i invariant by φ . Then there exists a relative metric ribbon graph (Γ , A ) (cid:44) → (Σ , ∂ Σ \ C ) and parametrizations of the cylinders (see Notation 4.4) r i : S × I → ˜Σ i such that:(i) φ (Γ) = Γ and the metric of Γ is also invariant by φ .(ii) l (˜Γ i ) = l ( C i ) .(iii) The projection from C i to ˜Γ i induced by r i is an isometry, that is, the map r ( θ, (cid:55)→ r ( θ, is an isometry.(iv) φ sends retractions lines (i.e. { θ } × I ) to retractions lines.Proof. The proof uses essentially the same technique used in the proof of [FPP,Theorems 5.14 and 5.27]. For completeness we outline it here.Let Σ φ be the orbit surface and suppose it has genus g and r ≥ k boundarycomponents. Let p : Σ → Σ φ be the induced branch cover.Take any relative spine Γ φ of Σ φ that:1) Contains all branch points of the map p .2) Contains the boundary components p ( ∂ Σ \ C ).3) Admits a metric such that p ( C i ) retracts to a part of the graph of length l ( C i ) /n .We observe that conditions 1) are 2) are trivial to get. Condition 3) followsbecause of the proof of in [FPP, Theorems 5.14 and 5.27]. There, the conditionson the metric of the graph Γ φ come from the rotation numbers of φ , however, wedo not use that these numbers come from φ in finding the appropriate graph soexactly the same argument applies.Observe that since the metric on C i is invariant by φ , there is a metric inducedon p ( C i ) for i = 1 , . . . k . Now choose any parametrizations (or product structures)of the cylinders in Σ φ Γ φ such that their retractions lines induce an isometry from p ( C i ) to ˜Γ φi .Define Γ := p − (Γ φ ). By construction, this graph satisfies ( i ) and ( ii ). Thepreimage by p of retraction lines on Σ φ gives rise to parametrizations of the cylindersin Σ Γ satisfying iii ) and iv ). (cid:3) Lemma 6.9.
Let (Γ • , B • , δ • ) be a relative mixed tˆete-`a-tˆete graph embedded in asurface Σ and let C , . . . , C k ⊂ B a set of relative boundary components cyclicallypermuted by φ . Suppose that all the vertices in these boundary components areof valency . Then we can modify the metric structure of the graph to producea mixed tˆete-`a-tˆete graph (ˆΓ • , ˆ B • , δ • ) with l ( C i ) as small as we want and with [ φ Γ ] ∂ Σ = [ φ ˆΓ ] ∂ Σ .Proof. Let e , . . . , e m be the edges comprising C , where e j has length l j . Let v , . . . , v m be the vertices of these edges, so that e i connects v i and v i +1 (here,indices are taken modulo m ). Let f i,j , for j = 1 , . . . , n i be the edges adjacentto v i , other than e i , e i +1 , in such a way that the edges have the cyclic order e i +1 , e i , f i, , . . . , f i,n i . Let ε < l ( C ). We would like to replace C with a circleof length l ( C ) − ε . We assume that l = min i l i .If ε/m ≤ l , then we do the following: (cid:96) Each edge e i is modified to have length l i − ε/m . (cid:96) For any i with n i = 1, the length of f i, is increased by ε/ m . (cid:96) For any i with n i >
1, extrude an edge g i from the vertex v i of length ε/ m so that one end of g i is adjacent to e i , e i +1 and g i , and the other isadjacent to f i, , . . . , f i,n i and g i , with these cyclic orders.In the case when ε/m > l , we execute the above procedure with ε replaced by m · l , which results in a circle made up of fewer edges. After finitely many steps,we obtain the desired length for C . IXED TˆETE-`A-TˆETE TWISTS AS MONODROMIES 15 l l l l ( f , ) l − l l − l l ( f , ) + l / l ( f , ) + l / l ( f , ) + l / (cid:15)/ Step 1 Step 2 l ( f , ) l ( f , ) v v v l ( f , ) + l / (cid:15)/ l ( f , ) + l / l ( f , ) + l / l − l − (cid:15)/ l − l − (cid:15)/ Figure 6.1.
Example of modification at a boundary component. Supposethat l < l < l . In step 1 we reduce the length of the circle by l . In step 2we reduce it by ε . (cid:3) Proof of Theorem 6.7.
By definition, all surfaces corresponding to vertices in D − (0)are connected because they are invariant by φ . We have that φ | Σ j, : Σ j, → Σ j, isperiodic outside a neighborhood of ∂ Σ ∩ Σ j, and that the fractional Dehn twistcoefficients with respect to all the components in ∂ Σ ∩ Σ j, are positive. Denote B j, := ∂ Σ j, \ ∂ Σ. By Theorem 4.6, for each j = 1 , . . . , β , there is a relative tˆete-`a-tˆete graph (Γ j, , B j, ) embedded in Σ j, modeling φ | Γ j, . Denote Γ[0] := (cid:70) j Γ j, and B [0] := (cid:70) j B j, . Then (Γ[0] , B [0] , δ ) is a relative mixed tˆete-`a-tˆete graph ofdepth 0 for Σ ≤ (it is just a relative tˆete-`a-tˆete graph) such that:(i) δ | Σ j, = π for all j = 1 , . . . , β .(ii) [ φ | Σ ] ∂ Σ = [ φ Γ[0] ] ∂ Σ .(iii) φ | B [0] = φ Γ[0] | B [0] .(iv) All the vertices on B [0] have valency 3.Suppose that we have a relative mixed tˆete-`a-tˆete graph (Γ[ a − • , B [ a − • , δ [ a − • ) of depth a embedded as a spine in Σ ≤ a and with B [ a −
1] = ∂ Σ ≤ a − \ ∂ Σsuch that:(i) δ [ a − i is a constant function for each i = 0 , . . . , a − φ | Σ ≤ a − ] ∂ Σ ≤ a − = [ φ Γ[ a − ] ∂ Σ ≤ a − (iii) φ | B [ a − = φ Γ[ a − | B [ a − .(iv) All the vertices on B [ a −
1] have valency 3.We recall that φ Γ[ a − denotes the mixed tˆete-`a-tˆete automorphism induced by(Γ[ a − • , B [ a − • , δ [ a − • ). We extend Γ[ a −
1] to a mixed tˆete-`a-tˆete graph Γ[ a ]satisfying (i) - (iv). This proves the theorem by induction. We focus on a particularorbit of surfaces. Fix j ∈ { , . . . , β a } and consider the surfaces Σ aj, , . . . , Σ aj,α j ⊂ Σ a with φ (Σ aj,k ) = Σ aj,k +1 and φ (Σ aj,α j ) = Σ aj, .For each j , we distinguish two types of boundary components in the orbit (cid:70) k Σ aj,k :Type I) Boundary components that are connected to an annulus whose other endis in Σ a − , we denote these by ∂ I .Type II) The rest: boundary components that are in ∂ Σ and boundary componentsthat are connected to an annulus whose other end is in Σ a +1 , we denotethese by ∂ II . Since we are doing the construction for an orbit, we use local notation in whichnot all the indices are specified so that the formulae is easier to read.Let A I denote the union of annuli connected to boundary components in ∂ I .These annuli are permuted by φ . Suppose that there are r (cid:48) different orbits of annuli A , . . . A r (cid:48) , and let (cid:96) i ∈ N be the length of the orbit A i . Let s i be the screw numberof the orbit A i (recall Definition 3.14 and Remark 3.15). Let B i, , . . . , B i,(cid:96) i be theorbit of boundary components of Σ a − that are contained in the orbit A i . Themetric of Γ[ a −
1] gives lengths to these boundary components and all the boundarycomponents in the same orbit have the same length l ( B i, ) ∈ R + . Consider thepositive real numbers(6.10) s (cid:96) l ( B , ) , . . . , s r (cid:48) (cid:96) r (cid:48) l ( B r (cid:48) , )Using Lemma 6.9, we modify the metric structure of Γ[ a −
1] near each orbit B i sothat s (cid:96) l ( B , ) = · · · = s r (cid:48) (cid:96) r (cid:48) l ( B r (cid:48) , ) . This is possible since we can make l ( B i, ) as small as needed.For each i = 1 , . . . , r (cid:48) , let A i, , . . . A i,(cid:96) i be the annuli in the orbit A i and let B (cid:48) i, , . . . B (cid:48) i,(cid:96) i be the boundary components that they share with Σ a . Considerparametrizations η i, , . . . , η (cid:96) i , given by Lemma 3.16. The metric on the bound-ary components of B [ a −
1] and the parametrizations induce a metric on all theboundary components in ∂ I that is invariant by φ .We observe that φ α j | Σ aj, : Σ aj, → Σ aj, is periodic and ∂ I ∩ Σ aj, is a subsetof boundary components that have a metric. So we can apply Lemma 6.8 andwe get a relative metric ribbon graph (Γ aj, , ∂ II ∩ Γ aj, ) and parametrizations ofeach cylinder in (Σ aj, ) Γ aj, with properties i ) , . . . , iv ) in the the Lemma. We cantranslate this construction by φ to the rest of the surfaces Σ aj, , . . . , Σ aj,α j . So weget graphs Γ aj,k (cid:44) → Σ aj,k and parametrizations for the cylinders in (Σ aj,k ) Γ aj,k for all k = 1 , . . . , α j . The construction assures us that φ | Σ aj,αj : Σ aj,α j → Σ aj, sends Γ aj,α j to Γ aj, isometrically and that it takes retractions lines of the parametrizations inΣ aj,α j to retraction lines in Σ aj, . We proceed to extend Γ[ a −
1] to the orbit of Σ aj, . For each i = 1 , . . . , r (cid:48) do thefollowing:Step 1. Remove B i, , . . . , B i,(cid:96) i from Γ[ a − ε > ε the metric on all the edges ofΓ[ a −
1] adjacent to vertices in B i, , . . . , B i,(cid:96) i .Step 3. Add to the graph the retraction lines of the parametrizations η i, , . . . , η i,(cid:96) i that were adjacent to vertices in B i, , . . . , B i,(cid:96) i . That is, if v ∈ B i, ⊂ A i, include η i, ( { v } × I ). Define the length of these segments as ε/ aj,k ) Γ aj,k that start at the ends of the lines added in the previousstep. Define the length of these segments as ε/ aj, , . . . , Γ aj,α j .We repeat this process for all orbits of surfaces in Σ a and so we extend the graphΓ[ a −
1] to all Σ a . Denote Γ[ a ] a := (cid:71) j,k Γ aj,k . IXED TˆETE-`A-TˆETE TWISTS AS MONODROMIES 17
Denote the resulting graph by Γ[ a ].We make the following observation: (Γ[ a ] Γ[ a ] a , ˜Γ[ a ] a ) is by construction isometricto (Γ[ a − , B [ a − φ Γ[ a ] Γ[ a ] a which acts on Σ ≤ a Γ[ a ] a . By the previous observation there is aninduced filtration on Γ[ a ]:Γ[ a ] = Γ[ a ] ⊃ Γ[ a ] ⊃ · · · ⊃ Γ[ a ] a − ⊃ Γ[ a ] a and similarly for the relative parts. We define δ a : Γ[ a ] a → R ≥ δ a and the parametrizations on the annuli that join Σ a − withΣ a we have that D δ a ◦ φ Γ Γ[ a ] : Σ Γ[ a ] a → Σ Γ[ a ] a is compatible with the gluing g a +1 . So that (Γ[ a ] , B [ a ]) is a relative mixed tˆete-`a-tˆete graph follows from I (cid:48) ) in Remark 5.7.We have already made sure in the construction that (i) and (iv) hold in Γ[ a ].Let’s show that (ii) and (iii) also hold. Observe that by construction φ leavesΓ[ a ] invariant so there is an automorphism ˜ φ a : Σ Γ[ a ] a → Σ Γ[ a ] a induced. This au-tomorphism coincides with D δ a ◦ φ Γ Γ[ a ] on ˜Γ[ a ] by the choice of the parametrizationsof the annuli A and by the choice of the number δ a . Also, by the choice of δ a and[FPP, Remark 7.28] we see that they have the same screw numbers on the annuliconnecting the level a − a . From this discussion we get (ii) and (iii)and finish the proof. (cid:3) Remark 6.11.
From the proof we get as an important consequence that a morerestrictive definition of a mixed tˆete-`a-tˆete graph is valid: it is enough to considermixed tˆete-`a-tˆete graphs where δ i is a constant function (i.e. a number) for all i = 0 , . . . , (cid:96) . Corollary 6.12.
The monodromy associated with a reduced holomorphic functiongerm defined on an isolated surface singularity is a mixed tˆete-`a-tˆete twist. Con-versely, let C (Γ) be the cone over the open book associated with a mixed tˆete-`a-tˆetegraph. Then there exists a complex structure on C (Γ) and a reduced holomorphicfunction germ f : C (Γ) → C inducing φ Γ as the monodromy of its Milnor fibration.Proof. The statement follows from Theorem 6.7 and [NP07, Theorem 2.1]. (cid:3)
Example 6.13.
Let Σ be the surface of Figure 6.2. Suppose it is embedded in R with its boundary component being the unit circle in the xy -plane. Consider therotation of π radians around the z -axis and denote it by R π . By the symmetricembedding of the surface, it leaves the surface invariant. Isotope the rotation sothat it is the identity on z ≤ /
2. We denote the isotoped automorphism by T . ˆ G ( φ ) G ( φ ) Σ Σ , Σ , Σ , A A A Figure 6.2.
On the left we see the surface Σ. On the right we see the corre-sponding graphs ˆ G ( φ ) and G ( φ ) for the depicted canonical form. More concretely, let T : R → R defined by(6.14) T ( r, θ, z ) := ( re i ( θ + π ) , z ) if z ≤ ε ( re θ + zε π , z ) if 0 ≤ z ≤ εid if z ≤ r, θ ) polar coordinates on the xy -plane and ε > D i be a full positive Dehn twist on the annuli A i , k = 1 , , φ := D − ◦ D − ◦ T | Σ . The automorphism comes already in canonical form. We construct the cor-responding Nielsen graph ˆ G ( φ ) and we observe that the corresponding distancefunction D is not a filtering function since there is 1 loop on ˆ G ( φ ). So we applyRemark 6.4 and we get the almost-canonical form and graphs of figure Figure 6.3. G ( φ, Σ) Σ Σ , Σ , Σ , Σ , ˆ G ( φ, Σ) Figure 6.3.
On the left we see the surface Σ. On the right we see the cor-responding graphs ˆ G ( φ ) and G ( φ ) for the depicted almost-canonical form. Inred we see the core curves of the annuli in A . IXED TˆETE-`A-TˆETE TWISTS AS MONODROMIES 19
Now there are 4 annuli in A in this almost-canonical form. The annuli A and A are exchanged by the monodromy, the Dehn twists D − and D indicate thatthe screw number of this orbit is −
1. And the annuli A , and A , that wereoriginally contained in A ; these annuli are also exchanged by the monodromy. Weget that the screw number on this orbit is − , B [0]) for φ | Σ : Σ → Σ . We use [FPP, Theorem 5.27] forthis. In Figure 6.4 we can see this graph in blue. π/ π/ π/ π/ π/ π/ π/ π/ π/ Figure 6.4.
The relative tˆete-`a-tˆete graph (Γ[0] , B [0]) embedded in Σ ⊂ Σ.The lengths are indicated on a few edges and the rest is obtained by symmetryof the graph. We can also see in red an invariant relative spine for Σ . In the next step we construct relative metric ribbon graphs for Σ . In this caseΣ consist of two connected surfaces that are exchanged. Each surface is a toruswith two disks removed and one of the boundary components is glued to an annulusconnecting it with Σ . This graph will correspond with Γ in the final mixed tˆete-`a-tˆete graph. In the notation of the Theorem we are using, it is Γ[1] . In Figure 6.4we can see these relative metric ribbon graphs in red. In this step we also choosean invariant product structure on Σ Now we proceed to find the parametrizations η and η . We pick any parametriza-tion η for A . On the right part of Figure 6.5 we can see the two retraction linesof the parametrization starting at the two vertices p and q of the correspondingboundary component in B [0] . On the left part of that figure we see two annuli, theupper one shows the image of the two retraction lines by φ , on the lower annuluswe see the retraction lines that we choose according to Lemma 3.16.We concatenate the chosen retraction lines of the annuli (green in 6.6) with thecorresponding retraction lines of the product structure in Σ (orange in the thepicture). A A A η ([0 , ×{ φ ( p ) } ) η ([0 , ×{ p } ) p φ ( p ) φ ( p ) q φ ( q ) φ ( q ) φ Figure 6.5.
The orbit of annuli A and A . On the right part we see A with a chosen product structure and on the left part we see two copies of A ,the lower one with the parametrization given by Lemma 3.16. The metric on the red part is chosen so that each of the two components of ˜Γ has the same length as the two relative components in B [0], that is equal to π/ − δ is the constant function π/ . π/ − (cid:15) π/ π/ π/ − (cid:15) A pq Figure 6.6.
The relative tˆete-`a-tˆete graph (Γ[1] , B [1]) embedded in Σ ⊂ Σ.The lengths are indicated. The green lines correspond to retraction lines of thecorresponding product structures η and η and the orange lines correspondto retraction lines of the product structures chosen for the cylinders Σ Similarly to the construction of the graph (Γ[1] , B [1]), we construct Γ[2] = Γ.We observe that Σ is an annulus whose boundary components are permuted by φ. It is attached along an orbit of annuli A , and A , to Σ . It has total lengthequal to 4 π/
72 = π/
18 Since this orbit of annuli has screw number − /
2, we getthat δ is the constant function π/ . IXED TˆETE-`A-TˆETE TWISTS AS MONODROMIES 21 π/ − (cid:15) π/ π/ π/ − (cid:15) A pq Figure 6.7.
The final mixed tˆete-`a-tˆete graph (Γ • , δ • ). In green we have Γ ;in red we have Γ \ Γ and in blue Γ \ Γ . References [A’C] Norbert A’Campo. Tˆete-`a-tˆete twists and geometric monodromy. Preprint, 2010.[A’C73] Norbert A’Campo. Sur la monodromie des singularit´es isol´ees d’hypersurfaces complexes.
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