aa r X i v : . [ m a t h . G T ] A p r ROOTS OF DEHN TWISTSABOUT SEPARATING CURVES
KASHYAP RAJEEVSARATHY
Abstract.
Let C be a curve in a closed orientable surface F of genus g ≥ F into subsurfaces e F i of genera g i , for i = 1 ,
2. Westudy the set of roots in Mod( F ) of the Dehn twist t C about C . All rootsarise from pairs of C n i -actions on the e F i , where n = lcm( n , n ) is thedegree of the root, that satisfy a certain compatibility condition. The C n i actions are of a kind that we call nestled actions, and we classifythem using tuples that we call data sets. The compatibility conditioncan be expressed by a simple formula, allowing a classification of allroots of t C by compatible pairs of data sets. We use these data set pairsto classify all roots for g = 2 and g = 3. We show that there is always aroot of degree at least 2 g + 2 g , while n ≤ g + 2 g . We also give someadditional applications. Introduction
Let F be a closed orientable surface of genus g ≥ C be a simpleclosed curve in F . Let t C denote a left handed Dehn twist about C .When C is a nonseparating curve, the existence of roots of t C is notso apparent. In their paper [7], D. Margalit and S. Schleimer showed theexistence of such roots by finding elegant examples of roots of t C whosedegree is 2 g + 1 on a surface of genus g + 1. This motivated an earliercollaborative work with D. McCullough [8] in which we derived necessaryand sufficient conditions for the existence of a root of degree n . As immediateapplications of the main theorem in the paper, we showed that roots of evendegree cannot exist and that n ≤ g +1. The latter shows that the Margalit-Schleimer roots achieve the maximum value of n among all the roots for agiven genus.Suppose that C is a curve that separates F into subsurfaces e F i of genera g i for i = 1 ,
2. It is evident that roots of t C exist. As a simple example,for the closed orientable surface of genus 2, we can obtain a square root ofthe Dehn twist t C by rotating one of the subsurfaces on either side of C byan angle π , producing a half-twist near C . As in the case for nonseparatingcurves, a natural question is whether we can give necessary and sufficientconditions for the existence of a degree n root of t C . In this paper, we derivesuch conditions and apply them to obtain information about the possible Date : June 4, 2018.
Key words and phrases. surface, mapping class, Dehn twist, separating curve, root. degrees. We use Thurston’s orbifold theory [12, Chapter 13] to prove themain result. A good reference for this theory is P.Scott [11].We start by defining a special class of C n -actions called nestled ( n, ℓ ) -actions . These C n -actions have a distinguished fixed point and the pointsfixed by some nontrivial element of C n form ℓ + 1 orbits. The equivalencyof two such actions will be given by the existence of a conjugating homeo-morphism that also satisfies an additional condition on their distinguishedfixed points. Two equivalence classes of actions will form a compatible pair ifthe turning angles of their representative actions around their distinguishedfixed points add up to 2 π/n . The key topological idea in our theory isdefining nestled ( n i , ℓ i )-actions on the subsurfaces ˜ F i for i = 1 , n = lcm( n , n ).Conversely, for each root of degree n , we reverse this argument to producea corresponding compatible pair.In Section 4, we introduce the abstract notion of a data set of degree n .As in the case of nonseparating curves, a data set of degree n is basically atuple that encodes the essential algebraic information required to describea nestled action. We show that equivalence classes of nestled ( n, ℓ )-actionsactually correspond to data sets, that is, each class has a correspondingdata set representation. Data sets D i of degree n i , for i = 1 , dataset pair ( D , D ) when they satisfy the formula nn k + nn k ≡ n ,where the turning angles at the centers of the disks are πk i n i mod 2 π . InTheorem 5.2, we show that this number-theoretic condition is an algebraicequivalent of the compatibility condition for actions, thus proving that dataset pairs correspond bijectively to conjugacy classes of roots. This theoremis essentially a translation of our topological theory of roots to the algebraiclanguage of data sets.As an immediate application of Theorem 5.2, we show the existence ofa root of degree lcm(4 g , g + 2), and in Section 6, we give calculationof roots in low-genus cases. In Section 7, we obtain some bounds on theorders of spherical nestled actions, that is, nestled actions whose quotientorbifolds are topologically spheres. For example, we prove that all nestled( n, ℓ )-actions for n ≥ (2 g −
1) have to be spherical. Finally, in Section 8,we use the main theorem and the results obtained in Section 7 to derivebounds on n . We show that in general, n ≤ g + 2 g and for any positiveinteger N , n ≤ g + (4 − N ) g + ( N − whenever both g i > N + 3.2. Nestled ( n, ℓ ) -actions An action of a group G on a topological space X is defined as a homomor-phism h : G → Homeo( X ). Since we are interested only in C n -actions, wewill fix a generator t for C n and identify the action with the isotopy class ofthe homeomorphism h ( t ) in Mod( X ). In this section, we introduce nestled( n, ℓ )-actions and give an example for such an action. These actions willplay a crucial role in the theory we will develop for roots of Dehn twists. OOTS OF DEHN TWISTS ABOUT SEPARATING CURVES 3
Definition 2.1.
An orientation-preserving C n -action on a surface F ofgenus at least 1 is said to be a nestled ( n, ℓ )-action if either n = 1, or n > C n form ℓ + 1 orbits.This is equivalent to the condition that the quotient orbifold has ℓ + 1 conepoints, one of which is a distinguished cone point of order n .A nestled ( n, ℓ )-action is said to be trivial if n = 1, that is, if it is theaction of the trivial group on F . In this case only, we allow a cone point oforder 1 in the quotient orbifold. The distinguished cone point can then beany point in F , and we require ℓ = 0. Definition 2.2.
Assume that F has a fixed orientation and fixed Riemann-ian metric. Let h be a nestled-( n, ℓ ) action on F with a distinguished fixedpoint P . The turning angle θ ( h ) for h is the angle of rotation of the inducedisomorphism h ∗ on the tangent space T P , in the direction of the chosenorientation. Example 2.3 (Margalit-Schleimer, [7]) . Rotate a regular (4 g + 2)-gon withopposite sides identified about its center P through an angle π ( g +1)(2 g +1) . Iden-tifying the opposite sides of P , we get a C g +1 -action h on S g with threefixed points denoted by P , x and y . Since the quotient orbifold has threecone points of order 2 g + 1, this defines a nestled (2 g + 1 , S g . If we choose P as the distinguished fixed point for the action h , then θ ( h ) = π ( g +1)(2 g +1) . P π xyx xy y Figure 1.
A nestled (2 g + 1 , g = 1. Remark 2.4.
Every nestled ( n, ℓ )-action has an invariant disk around itsdistinguished fixed point. Let F be a closed oriented surface with a fixed KASHYAP RAJEEVSARATHY
Riemannian metric ρ , and let h be a nestled ( n, ℓ )-action on F with a dis-tinguished fixed point P . Consider the Riemannian metric ¯ ρ defined by h v, w i ¯ ρ = 1 n n X i =1 h h i ∗ ( v ) , h i ∗ ( w ) i ρ , where v, w ∈ T P F . Under this metric ¯ ρ , h is an isometry. Since there exists ǫ > exp P : B ǫ (0) ⊂ T P F → B ǫ ( P ) ⊂ F is a diffeomorphism, h preserves the disk B ǫ ( P ). Definition 2.5.
Two nestled ( n, ℓ )-actions h and h ′ on F with distinguishedfixed points P and P ′ are equivalent if there exists an orientation-preservinghomeomorphism t : F → F such that(i) t ( P ) = P ′ .(ii) tht − is isotopic to h ′ relative to P ′ . Remark 2.6.
By definition, equivalent nestled ( n, ℓ )-actions h and h ′ on F are conjugate in Mod( F ). Since conjugate homeomorphisms have the samefixed point data, we have that θ ( h ) = θ ( h ′ ).3. Compatible pairs and roots
Suppose that C is a curve that separates a surface F of genus g into twosubsurfaces. As mentioned earlier, the central idea is defining compatiblenestled actions on the subsurfaces that “fit together” to give a degree n rootof the Dehn twist t C . We will show in Theorem 3.4 that compatible pairs ofequivalent actions correspond bijectively to conjugacy classes of roots of t C . Notation 3.1.
Suppose that C separates a closed orientable surface F intosubsurfaces of genera g and g , where g ≥ g . Let F i denote the closedsurface obtained by coning the subsurface of genus g i . We will think of F as ( F , C ) F , C ), that is, the surface obtained by taking the connectedsum of the F i along C . For the sake of convenience, we will denote this by F = F C F . Definition 3.2.
Equivalence classes [ h i ] of nestled ( n i , ℓ i )-actions h i onclosed oriented surfaces F i for i = 1 , compatible pair ([ h ] , [ h ]) if θ ( h ) + θ ( h ) = 2 π/n mod 2 π .The integer n = lcm ( n , n ) is called the degree of the compatible pair.We may treat ([ h ] , [ h ]) to as an unordered pair, since ([ h ] , [ h ]) is a com-patible pair if and only if ([ h ] , [ h ]) is. Lemma 3.3.
Let F be a compact orientable surface, possibly disconnected.If h : F → F is a homeomorphism such that h n is isotopic to id F , then h isisotopic to a homeomorphism j with j n = id F .Proof. When F is connected, this is the Nielsen-Kerchkoff theorem [4, 5, 9].Suppose that F is not connected. We may asssume that h acts transitivelyon the set of components F , F , ..., F ℓ of F . Choose notation so that h | F i : OOTS OF DEHN TWISTS ABOUT SEPARATING CURVES 5 F i → F i +1 and h | F ℓ − : F ℓ − → F . Since h n = ( h l ) n/ℓ ≃ id F , the Nielsen-Kerchkoff theorem implies that h ℓ | F ≃ j where j is a homeomorphism on F with j n/ℓ = id F . Therefore, id F ≃ j ◦ ( h ℓ | F ) − via an isotopy K t .Define an isotopy H t of h by H t | F i = h for 1 ≤ i ≤ ℓ − H t | F ℓ − = K t ◦ h | F ℓ − . Then, H | F ℓ − = K ◦ h = j ◦ ( h ℓ | F ) − ◦ h . We see that( H | F i ) ℓ = h i ◦ ( j ◦ h − ℓ ) ◦ h ℓ − − i = h i ◦ j ◦ h − i and ( H | F i ) n = ( H | F i ℓ ) n/ℓ = h i ◦ j n/ℓ ◦ h − i = h i ◦ h − i = id F i . The required homeomorphism is j = H . (cid:3) Theorem 3.4.
Let F = F C F be a closed oriented surface of genus g ≥ .Then the conjugacy classes in Mod( F ) of roots of t C of degree n correspondto the compatible pairs ([ h ] , [ h ]) of equivalence classes of nestled ( n i , ℓ i ) -actions h i on F i of degree n .Proof. We will first prove that every root of degree n yields a compatiblepair of ([ h ] , [ h ]) of degree n .Fix a closed annulus neighborhood N of C . Let e F i for i = 1 , G − N , and denote the genus of e F i by g i . We fix coordinateson F so that the subsurface e F is to the left of C as shown in Figure 2. Byisotopy we may assume that t C ( C ) = C , t C ( N ) = N , and t C | e F i = id e F i for i = 1 , C e F e F N Figure 2.
The surface F with the separating curve C andthe tubular neighborhood N of C .Suppose that h is an n th root of t C . We have t C ≃ ht C h − ≃ t h ( C ) , whichimplies that h ( C ) is isotopic to C . Changing h by isotopy, we may assumethat h preserves C and takes N to N . Put e h i = h | f F i for i = 1 ,
2. Since h n ≃ t C and both preserve C , there is an isotopy from h n to t C preserving C and hence one taking N to N at each time. That is, f h n is isotopicto id e F and f h n is isotopic to id e F . By Lemma 3.3, e h i is isotopic to ahomeomorphism whose n th power is id e F i for i = 1 ,
2. So we may change e h i and hence h by isotopy to assume that e h in = id f F i for i = 1 , n i be the smallest positive integer such that e h in i = id f F i for i = 1 , s = lcm ( n , n ). Clearly, s | n since n i | n . Also, h s = id e F ∪ e F which KASHYAP RAJEEVSARATHY implies that h s = t C d for some integer d . Hence, ( h s ) n/s = ( t C d ) n/s i.e. h n = t C dn/s . We get, t C = t C dn/s which implies that dn/s = 1 since nohigher power of t C is isotopic to t C . Hence, d = 1 and n = s = lcm ( n , n ).Assume for now that h does not interchange the sides of C . We fill inthe boundary circles of e F and e F with disks to obtain the closed orientablesurfaces F and F with genera g and g . We then extend e h i to a home-omorphism h i on F i by coning. Thus h i defines a C n i action on F i where n i | n , C n i = h h i | h n i i = 1 i for i = 1 , lcm ( n , n ) = n . Since the home-omorphism h i fixes the center point P i of the disk F i − e F i , we choose P i asthe distinguished fixed point for h i . So h i defines a nestled ( n i , ℓ i )-action on F i for some ℓ i .The orientation on F restricts to orientations on the F i , so that we mayspeak of rotation angles θ ( h i ) for h i . Then the rotation angle θ ( h i ) = 2 πk i /n i for some k i with gcd( k i , n i ) = 1. As seen in Figure 3, the difference inturning angles equals 2 πk /n − ( − πk /n ) = 2 π/n , giving θ ( h ) + θ ( h ) ≡ π/n mod 2 π . That is, ( h , h ) is a compatible pair. AP h ( A ) Bh ( B ) P AB Figure 3.
The local effect of h and h on disk neighbor-hoods of P and P in F and F , and the effect of h onthe neighborhood N of C in F . Only the boundaries of thedisk neighborhoods are contained in e F i , where they form theboundary of N . The rotation angle θ ( h ) is 2 πk /n and theangle θ ( h ) is 2 πk /n = 2 π (1 /n − k /n ).Suppose now that h interchanges the sides of C . Then h must be of evenorder, say 2 n , and h preserves the sides of C and is of order n . Since theactions of h | f F i on the e F i are conjugate by h | f F ∪ f F , these actions will induceconjugate C n -actions on the coned surfaces F i . Consequently, these inducedactions will have the same turning angles at the centers P i of the coneddisks of F i . For this compatible pair of nestled ( n i , ℓ i )-actions, say ( h , h ),associated with h , we must have θ ( h ) = θ ( h ) = π/n and n = n = n .If we extend to N using a simple left-handed twist, the twisting angle is2 πk/n , and consequently h n = t kC . Other extensions will differ from thisby full twists, giving h n = t k +2 jnC for some integer j . In any case, h n cannot equal t C . This proves that h cannot reverse the sides of C . OOTS OF DEHN TWISTS ABOUT SEPARATING CURVES 7
Suppose that we have roots h and h ′ that are conjugate in Mod( F ), thatis, there exists t ∈ Mod( F ) such that h ′ = t ◦ h ◦ t − . Then ( h ′ ) n = t ◦ h n ◦ t − ,that is, t C = t ◦ t C ◦ t − = t t ( C ) . This shows that C and t ( C ) are isotopiccurves. Changing t by isotopy, we may assume that t ( C ) = C and t ( N ) = N .Let t i , h i and h ′ i respectively denote the extensions of t | f F i , h | f F i and h ′ | f F i to F i by coning.Assume for now that t does not exchange the sides of C . Since t , h and h ′ all preserve N , we may assume that the isotopy from t ◦ h ◦ t − to h ′ preserves N , and consequently each t i ◦ h i ◦ t i − is isotopic to h ′ i preserving P i . Since t i takes P i to P i , h i and h ′ i are equivalent as nestled ( n i , ℓ i )-actionson F i , so h and h ′ produce the same compatible pair ([ h ] , [ h ]).Suppose that t exchanges the sides of C . Then g = g , h ′ − i ≃ t i ◦ h i ◦ t i − and t i ( P i ) = P − i . So the actions h and h ′ are equivalent, as are actions h ′ and h . Therefore, the (unordered) compatible pairs for the two rootsare the same.Conversely, given a compatible pair ([ h ] , [ h ]) of equivalence classes ofnestled ( n i , ℓ i )-actions, we can reverse the argument to produce a root h .For let P i denote the distinguished fixed point of h i and let p i denote thecorresponding cone point of order n i in the quotient orbifold O i . By Re-mark 2.4, there exists an invariant disk D i for h i around p i . Removing D i produces the surfaces e F i , and attaching an annulus N produces the surface F of genus g . Condition (ii) on compatible pairs ensures that the rotationangles work correctly to allow an extension of h | e F ∪ h | e F to an h with h n being a single Dehn twist about C .It remains to show that the resulting root h of t C is determined up toconjugacy in the mapping class group of F . Suppose that h ′ i ∈ [ h i ]. Let P ′ i denote the distinguished fixed point for h ′ i , and let D ′ i be an invariant diskfor h ′ i around P ′ i . Removing the D ′ i s produces surfaces e F ′ i ∼ = F i , for i = 1 , N ′ with a 1 /n th twist, extends h ′ | e F ′ ∪ h ′ | e F ′ toa homeomorphism h ′ on a surface F ′ ∼ = F of genus g . Since h ′ i ∈ [ h i ], bydefinition, there exists t i such that t i ( P i ) = P ′ i and t i ◦ h i ◦ t i − ≃ h ′ i rel P ′ i via an isotopy H i in Mod( F ′ i ). Since h i and h ′ i have finite order andare conjugate up to isotopy by t i , we may assume that t i ( D i ) = D ′ i and,identifying F and F ′ using t , that the isotopy H i from t i ◦ h i ◦ t i − to h ′ i isrelative to D i . With respect to this identification, we choose a k : N → N such that h ′ | N = k ◦ h | N ◦ k − . Now define t : F → F by t | f F i = h i | f F i , and t | N = k . Then h ′ ≃ t ◦ h ◦ t − via an isotopy H given by H | f F i = H i | f F i , and H | N = id N . (cid:3) Nestled ( n, ℓ ) -actions and data sets In this section, we introduce the language of data sets of degree n in orderto algebraically encode classes of nestled ( n, ℓ )-actions. We will also prove KASHYAP RAJEEVSARATHY that equivalence classes of nestled ( n, ℓ )-actions actually correspond to datasets.
Definition 4.1. A data set for F is a tuple D = ( n, e g, a ; ( c , x ) , . . . , ( c ℓ , x ℓ ))where n , e g and the x i are integers, a is a residue class modulo n , and each c i is a residue class modulo x i , such that(i) n ≥ e g ≥
0, each x i >
1, and each x i divides n .(ii) gcd( a, n ) = gcd( c i , x i ) = 1.(iii) a + ℓ X i =1 nx i c i ≡ n .The number n is called the degree of the data set. If n = 1, then we requirethat a = 1, and the data set is D = (1 , e g,
1; ). The integer g defined by g = e gn + 12 (1 − n ) + 12 ℓ X i =1 nx i ( x i − genus of the data set. We consider two data sets to be the sameif they differ by reordering the pairs ( c , x ) , . . . , ( c ℓ , x ℓ ). Remark 4.2.
For any data set D = ( n, e g, a ; ( c , x ) , . . . , ( c ℓ , x ℓ )),lcm { x , x , . . . , x n } = n . To see this, put k = lcm ( x , x , . . . , x ℓ ). Since each x i | n , k | n . So it remains to show that n | k . Condition ( iii ) implies that akk + ℓ X i =1 n ( k/x i ) k c i ≡ n . Multiplying by k we get ak + n ℓ X i =1 ( k/x i ) c i ≡ n . Since gcd ( a, n ) = 1, we have n | k .We will prove in the following proposition that data sets of degree n correspond to equivalence classes nestled-( n, ℓ ) actions. Proposition 4.3.
Data sets of degree n and genus g correspond to equiva-lence classes of nestled ( n, ℓ ) -actions on closed orientable surfaces of genus g .Proof. Let h be a nestled-( n, ℓ ) action on a closed orientable surface F ofgenus g . Let O be the quotient orbifold for the action and let e g be the genusof its underlying 2-manifold. Let P be the distinguished fixed point of h andlet p be the cone point in O of order n that is its image in O . Let p , . . . , p ℓ be the other possible cone points of O , if any.Figure 4 shows a generator α of the orbifold fundamental group π orb ( O )that goes around the point p , and generators γ i , ≤ i ≤ ℓ going around p i .Let a i and b i , 1 ≤ j ≤ e g be standard generators of the “surface part” of O ,chosen to give the following presentation of π orb ( O ): OOTS OF DEHN TWISTS ABOUT SEPARATING CURVES 9 pp p αγ γ Figure 4.
The orbifold O π orb ( O i ) = h α, γ , . . . , γ ℓ , a , b , . . . , a e g , b e g | α n = γ x = · · · = γ x i i = 1 , αγ · · · γ ℓ = e g Y [ a i , b i ] i . From orbifold covering space theory [12], we have the following exactsequence: 1 −→ π ( F ) −→ π orb ( O ) ρ −→ C n −→ . The homomorphism ρ is obtained by lifting path representatives of elementsof π orb ( O )— these do not pass through the cone points so the lifts areuniquely determined.For 1 ≤ i ≤ l , the preimage of p i consists of n/x i points cyclically per-muted by h , where x i is the order of the stabilizer of each point in thepreimage of p i . Each of the points has stabilizer generated by h n/x i . Itsrotation angles must be the same at all points of the orbit, since its actionat one point is conjugate by a power of h to its action at each other point.So the rotation angle at each point is of the form 2 πc ′ i /x i , where c ′ i is aresidue class modulo x i and gcd( c ′ i , x i ) = 1. Lifting the γ i , we have that ρ ( γ i ) = h ( n/x i ) c i where c i c ′ i ≡ x i .Finally, we have ρ ( Q e gi =1 [ a i , b i ]) = 1, since C n is abelian, so1 = ρ i ( αγ · · · γ ℓ ) = t a +( n/x ) c + ··· +( n/x i ) c i giving a + ℓ X i =1 nx i c i ≡ n . The fact that the data set D has genus equal to g follows easily from themultiplicativity of the orbifold Euler characteristic for the orbifold covering F → O :(4.1) 2 − gn = 2 − e g + (cid:18) n − (cid:19) + ℓ X i =1 (cid:18) x i − (cid:19) Thus, h gives a data set D = ( n, e g, a ; ( c , x ) , . . . , ( c ℓ , x ℓ )) of degree n andgenus g . Consider another nestled ( n, ℓ )-action h ′ in the equivalence class of h witha distinguished fixed point P ′ . Then by definition there exists an orientation-preserving homeomorphism t ∈ Mod( F ) such that t ( P ) = P ′ and th ′ t − isisotopic to h relative to P . Therefore, the two actions will have the samefixed point data and hence produce the same data set D .Conversely, given a data D = ( n, e g, a ; ( c , x ) , . . . , ( c ℓ , x ℓ )), we can reversethe argument to produce an equivalence class of a nestled ( n, ℓ )-action h on a surface F of genus g . We construct the orbifold O and representation ρ : π orb ( O ) → C n . Any finite subgroup of π orb ( O ) is conjugate to one of thecyclic subgroups generated by α or a γ i , so condition (ii) in the definition ofthe data set ensures that the kernel of ρ is torsionfree. Therefore the orbifoldcovering F → O corresponding to the kernel is a manifold, and calculationof the Euler characteristic shows that F has genus g .It remains to show that the resulting action on F is determined up toour equivalence in Mod( F ). Suppose that two actions h and h ′ on F withdistinguished fixed points P and P ′ have the same data set D . D encodesthe fixed-point data of the periodic transformations h . By a result of J.Nielsen [9] (see also A. Edmonds [2, Theorem 1.3]), h and h ′ have to beconjugate by an orientation-preserving homeomorphism t . As in the proofof Theorem 1.1 in [8], t may be chosen so that it preserves t ( P ) = P ′ . Thus D determines h up to equivalence. (cid:3) Proposition 4.3 enables us to view equivalence classes of nestled ( n, ℓ )-actions simply as data sets.
Notation 4.4.
We will denote a data set of degree n and genus g by D n,g,i ,where i is an index. The trivial data set D = { , g, } , for any g , will bedenoted by D ,g . Example 4.5.
For every g ≥
1, below are examples of data sets that rep-resent nestled ( n, n is 2 g + 1, 4 g and 4 g + 2:(i) D g +1 ,g, = (2 g + 1 , ,
1; ( g, g + 1) , ( g, g + 1)).(ii) D g,g, = (4 g, ,
1; (1 , , (2 g − , g )).(iii) D g +2 ,g, = (4 g + 2 , ,
1; (1 , , ( g, g + 1)). Remark 4.6.
For the data set D = ( n, e g, a ; ( c , x ) , . . . , ( c n , x ℓ )) associatedwith a nestled ( n, ℓ )-action, Equation 4.1 in the proof of Proposition 4.3gives the following inequality(4.2) 1 − gn = − ( ℓ − − e g + ℓ X i =1 x i ≤ − ( ℓ −
1) + ℓ X i =1 x i . Let O be the quotient orbifold for a nestled ( n, ℓ )-action. Let α be agenerator of O going around the distinguished order n cone point and let γ , γ , . . . , γ ℓ be generators going around the other cone points. We have theexact sequence 1 −→ π ( F ) −→ π orb ( O ) ρ −→ C n −→ . OOTS OF DEHN TWISTS ABOUT SEPARATING CURVES 11
Remark 4.7.
There exists no non-trivial action with ℓ = 0. Suppose thatwe assume the contrary. Then O has a distinguished cone point of order n and no other cone points. Let a j and b j , 1 ≤ j ≤ e g be the standardgenerators of the “surface part” of O . Then, the fundamental group of O has the following representation π orb ( O ) = h α, a , b , . . . , a e g , b e g | α n = 1 , α = e g Y j =1 [ a j , b j ] i . Since C n is abelian, ρ ( α ) = ρ ( Q e gj =1 [ a j , b j ]) = 1, which is impossible since ρ has torsion free kernel.5. Data set pairs and roots
By Theorem 3.4, each conjugacy class of a root of t C in Mod( F ) corre-sponds to a compatible pair ([ h ] , [ h ]) of (equivalence classes of) nestledactions, and by Proposition 4.3, such a pair determines a pair ( D , D ) ofdata sets. To determines which pairs arise, we must replace the geomet-ric compatibility condition in Theorem 3.4 by an algebraic compatibilitycondition on the corresponding data sets. Definition 5.1.
Two data sets D = ( n , e g , a ; ( c , x ) , . . . , ( c ℓ , x ℓ )) and D = ( n , e g , a ; ( c , x ) , . . . , ( c m , x m )) are said to form a data set pair ( D , D ) if(5.1) nn k + nn k ≡ n where n = lcm( n , n ) and a i k i ≡ n i . Note that although the k i areonly defined modulo n i , the expressions nn i k i are well-defined modulo n . Theinteger n is called the degree of the data set pair and g = g + g is calledthe genus of the data set pair. We consider ( D , D ) to be an unorderedpair, that is, ( D , D ) and ( D , D ) are equivalent as compatible pairs.We can now reformulate Theorem 3.4 in terms of data sets. Theorem 5.2.
Let F = F C F be a closed oriented surface of genus g ≥ .Then, data set pairs ( D , D ) of degree n and genus g , where D is a dataset of genus g and D is a data set of genus g , correspond to the conjugacyclasses in M od ( F ) of roots of t C of degree n .Proof. Let h denote the conjugacy class of a root of t C of degree n withcompatible pair representation ([ h ] , [ h ]). From Proposition 4.3, the h i cor-respond to data sets D i = ( n i , e g i , a i ; ( c i , x i ) , . . . , ( c iℓ i , x ℓ i )). So it sufficesto show that the geometric condition θ ( h ) + θ ( h ) = 2 π/n in Definition 3.2is equivalent to the condition nn k + nn k ≡ n in Definition 5.1.As in the proof of Proposition 3.4, let P i denote the center of the fillingdisk of the subsurface e F i of genus g i . Choosing P i as the distinguished fixedpoint of h i , we get that θ ( h i ) = 2 πk i /n i , where gcd( k i , n i ) = 1 and a i k i ≡ n i . Since h n = t C , the left-hand twisting angle along N is 2 π/n ,which equals 2 πk /n − ( − πk /n ) = 2 π/n , giving nn k + nn k ≡ n .The converse is just a matter of reversing the argument. (cid:3) Corollary 5.3.
Suppose that F = F C F . Then there always exists a rootof the Dehn twist t C about C of degree lcm (4 g , g + 2) .Proof. As in Theorem 5.2, let e F i denote the subsurfaces obtained by cut-ting F along C , and let F i denote the surfaces obtained by adding disksto the F i . Let n = 4 g and n = 4 g + 2. From Example 4.5, forany residue class a i modulo n i with gcd( a i , n i ) = 1, the data set D =( n , , a ; ( − a , g ) , ( a , g )) defines a nestled ( n , F of genus g , and the data set D = ( n , , a ; ( a , , ( a g , g + 1)) de-fines a nestled ( n , F of genus g .Let k i denote the inverse of a i modulo n i and let n = lcm( n , n ). We willnow show that the a i can be selected so that Equation 5.1 is satisfied. Inother words, this will prove that D and D form a data set pair ( D , D ).Since nn and nn are relatively prime, there always exist integers p and q such that nn p + nn q = 1 . In particular, since nn and nn are not both odd, by [8, Lemma 7.1], p and q can be chosen so that gcd ( p, n ) = gcd ( q, n ) = 1. Let k be the residueclass of p modulo n and let k be the residue class of q modulo n . Takingmodulo n , we get nn k + nn k ≡ n . Therefore, by Theorem 5.2, there exists a root of t C of order lcm (4 g , g +2). (cid:3) Corollary 5.4.
Let F = F C F be a closed oriented surface of genus g ≥ . Suppose that M denotes the maximum degree of a root of the Dehntwist t C about C . Then g + 2 g ≤ M .Proof. If g is even, then Corollary 5.3 with g = g = g gives a root ofdegree lcm(2 g, g + 1) = 2 g (2 g + 1). If g is odd, then g = g +12 and g = g − gives a root of degree lcm(2( g + 1) , g ) ≥ g ( g + 1). (cid:3) Classification of roots for the closed orientable surfacesof genus 2 and 3
Surface of genus 2.
Let F denote the closed orientable surface ofgenus 2. Up to homeomorphism, there is a unique curve C that separates F into two subsurfaces of genus 1. Given a root of t C , the process describedin the proof of Theorem 5.2 produces orientation-preserving C n i actions onthe tori F i for i = 1 , n = lcm ( n , n ).If a cyclic group C n acts faithfully on a surface F fixing a point x ,then the map C n −→ Aut ( π ( F, x )) is a monomorphism [1, Theorem 2, OOTS OF DEHN TWISTS ABOUT SEPARATING CURVES 13 p.43]. We also know that the group of orientation-preserving automorphisms
Aut + ( π ( F i , x )) ∼ = Aut + ( Z × Z ) ∼ = SL (2 , Z ) ∼ = Z ∗ Z Z . Since any elementof finite order of an amalgamated product A ∗ C B is conjugate into one ofthe groups A or B [6], it can only be of order 2, 3, 4 or 6. Taking theleast common multiple of any two of these orders gives 12 as the only otherpossibility for the order of a root of t C . We summarize these inferences inthe following corollary. Corollary 6.1.
Let F be the closed orientable surface of genus 2 and C aseparating curve in F . Then a root of a Dehn twist t C about C can only beof degree 2, 3, 4, 6, or 12. Given below are the data set pairs that represent each conjugacy class ofroots.For n = 2:(i) ( D , , , D , ), where D , , = (2 , ,
1; (1 , , (1 , , (1 , n = 3:(i) ( D , , , D , ), where D , , = (3 , ,
1; (1 , , (1 , D , , , D , , ), where D , , = (3 , ,
2; (2 , , (2 , n = 4:(i) ( D , , , D , ), where D , , = (4 , ,
1; (1 , , (1 , D , , , D , , ), where D , , = (4 , ,
3; (1 , , (3 , n = 6:(i) ( D , , , D , ), where D , , = (6 , ,
1; (1 , , (1 , D , , , D , , ), where D , , = (6 , ,
5; (1 , , (2 , D , , , D , , ).For n = 12:(i) ( D , , , D , , ).(ii) ( D , , , D , , ).It can be shown using elementary calculations that these are the only pos-sible roots for the various orders. For example, when n = 12, the conditionlcm( n , n ) = 12 would imply that the set { n , n } can be either { , } or { , } . When n = 6 and n = 4, the data set pair condition gives2 k + 3 k ≡ k i is a residue modulo n i , the only possiblesolution to this equation is k = 5 and k = 1. This would imply that a = 5and a = 1 since a i is the inverse of k i modulo n i . Geometrically, this rep-resents the root h of t C whose twisting angle on one side is 2 πk /n = 5 π/ C is 2 πk /n = π/
2. Each data set D i in the dataset pair ( D , D ) is then uniquely determined by condition ( iii ) (for datasets) and the formula for calculating the genus g i . Similar calculations canbe used to determine all the data set pairs for the surface of genus 3.6.2. Surface of genus 3.
Up to homeomorphism, the surface of genus g = 3 has a unique curve that separates the surface into two subsurfaces ofgenera 2 and 1. Given below are the data set pairs that represent roots of various degrees.For n = 2:(i) ( D , , D , , ).(ii) ( D , , , D , ), where D , , = (2 , ,
1; (1 , , (1 , , (1 , , (1 , , (1 , D , , , D , ), where D , , = (2 , ,
1; (1 , n = 3:(i) ( D , , D , , ).(ii) ( D , , , D , ), where D , , = (3 , ,
1; (2 , , (2 , , (1 , D , , , D , ), where D , , = (3 , ,
2; (1 , , (1 , , (2 , n = 4:(i) ( D , , D , , ).(ii) ( D , , , D , ), where D , , = (4 , ,
1; (1 , , (1 , , (3 , D , , , D , , ), where D , , = (4 , ,
3; (1 , , (1 , , (2 , n = 5:(i) ( D , , , D , ), where D , , = (5 , ,
1; (1 , , (3 , D , , , D , ), where D , , = (5 , ,
1; (2 , , (2 , n = 6:(i) ( D , , D , , ).(ii) ( D , , , D , ), where D , , = (6 , ,
1; (2 , , (1 , D , , , D , , ).(iv) ( D , , , D , , ).(v) ( D , , , D , , ).(vi) ( D , , , D , , ).(vii) ( D , , , D , , ), where D , , = (6 , ,
5; (1 , , (5 , n = 8:(i) ( D , , , D , ), where D , , = (8 , ,
1; (1 , , (3 , D , , , D , , ), where D , , = (8 , ,
5; (1 , , (7 , D , , , D , , ), where D , , = (8 , ,
7; (1 , , (5 , D , , , D , , ), where D , , = (8 , ,
3; (1 , , (1 , n = 10:(i) ( D , , , D , ), where D , , = (10 , ,
1; (1 , , (2 , D , , , D , , ), where D , , = (5 , ,
3; (1 , , (1 , D , , , D , , ), where D , , = (5 , ,
3; (3 , , (4 , n = 12:(i) ( D , , , D , , ).(ii) ( D , , , D , , ).(iii) ( D , , , D , , ).(iv) ( D , , , D , , ).For n = 15:(i) ( D , , , D , , ), where D , , = (5 , ,
3; (1 , , (1 , D , , , D , , ), where D , , = (5 , ,
3; (3 , , (4 , n = 20: OOTS OF DEHN TWISTS ABOUT SEPARATING CURVES 15 (i) ( D , , , D , , ), where D , , = (5 , ,
4; (4 , , (2 , D , , , D , , ), where D , , = (5 , ,
4; (3 , , (3 , D , , , D , , ), where D , , = (10 , ,
7; (1 , , (4 , n = 24:(i) ( D , , , D , , ).(ii) ( D , , , D , , ).For n = 30:(i) ( D , , , D , , ), where D , , = (10 , ,
9; (1 , , (3 , D , , , D , , ), where D , , = (5 , ,
1; (1 , , (3 , D , , , D , , ), where D , , = (5 , ,
1; (2 , , (2 , Spherical nestled actions
A spherical action is simply a nestled ( n, ℓ )-action whose quotient orbifoldis a sphere. We will show in Proposition 7.3 that nestled ( n, ℓ )-actions mustbe spherical when n is sufficiently large. This means that in order to derivebounds on n , it suffices to restrict attention to spherical actions. We willalso derive several other results on spherical actions which we will be helpfulin later sections. Definition 7.1.
A non-trivial nestled ( n, ℓ )-action is said to be spherical ifthe underlying manifold of its quotient orbifold is topologically a sphere.
Example 7.2.
The actions in Examples 1 and 4.5 are spherical actions.
Proposition 7.3. If n > (2 g − , then every nestled ( n, ℓ ) -action on F is spherical.Proof. Let D = ( n, e g, a ; ( c , x ) , . . . , ( c n , x ℓ )) be the data set associated witha nestled ( n, ℓ )-action on F . Equation 4.2 gives(7.1) e g = 12 + 2 g − n − ℓ ℓ X i =1 x i , Each x i ≥
2, and by Remark 4.7, we must have ℓ ≥
1, so this becomes e g ≤
12 + 2 g − n − ℓ ≤
14 + 2 g − n . That is, e g ≥ n ≤ (4 g − / (cid:3) Remark 7.4.
There exists no spherical nestled ( n, ℓ )-action with ℓ = 1.Suppose we assume on the contrary that ℓ = 1. Then, Equation 4.1 wouldimply that 1 − gn = 1 x . This is impossible since x > g ≥ Proposition 7.5.
Suppose that a surface F of genus g has a spherical nes-tled ( n, ℓ ) -action. Write the prime factorization of n as n = p a q a · · · q ka k where p a > q ia i for each i ≥ , and write q for min { p, q , . . . , q k } . If n > g − − q − p a , then ℓ = 2 .Proof. Each x i ≥ q , and by Proposition 4.2, at least one x i ≥ p a . UsingEquation 7.1 we have0 = 12 + 2 g − n − ℓ ℓ X i =1 x i ≤
12 + 12 p a + 2 g − n − ℓ ℓ − qℓ ≤ q ( q − p a + qq − (cid:18) g − n (cid:19) The right-hand side of the latter inequality is less than 3 when the inequalityin the proposition holds. Therefore, by Remark 7.4, ℓ = 2. (cid:3) Corollary 7.6.
Suppose that a surface F of genus g has a spherical nestled ( n, ℓ ) -action. (i) If n = 2 , then ℓ = 2 g + 1 . In particular, there does not exist aspherical nestled (2 , -action. (ii) If n = 3 , then ℓ = g + 1 . There exists a spherical nestled (3 , -actionif and only if g = 1 . (iii) If n is even, n ≥ , and n > (2 g − , then ℓ = 2 . (iv) If n is odd, n ≥ , and n > (2 g − , then ℓ = 2 .Proof. For (i), an Euler characteristic calculation shows that ℓ = 2 g + 1when n = 2. These are exactly the hyperelliptic actions.For (ii), when n = 3, an Euler characteristic calculation shows that ℓ = g + 1, and as seen in Section 6, there is a nestled (3 , n = 6. In Proposition 7.5 we have q = 2and p a = 3, giving the conclusion that if 6 > (2 g − ℓ = 2. Thecondition 6 > (2 g −
1) holds exactly when g ≤
2, so (iii) is true in thiscase. One can check that there exist nestled (6 , g ≤
2. For the cases of (iii) other than n = 6, we have q = 2 and p a ≥ q ≥ p a ≥
5. Again Proposition 7.5 gives theresult. (cid:3) Bounds on the degree of a root
In this section, we use the Theorem 5.2 and the results derived in Sec-tions 2 and 7 to derive some results on the degree n of a root. Among theresults derived is an upper bound and a stable upper bound for n . OOTS OF DEHN TWISTS ABOUT SEPARATING CURVES 17
Remark 8.1.
It is a well known fact [3] that the maximum order for anautomorphism of a surface of genus g is 4 g + 2. In Example 4.5, we showedthat a nestled action of order 4 g + 2 always exists. Proposition 8.2.
There exists no nestled (4 g + 1 , ℓ ) -action.Proof. By Proposition 7.3, a nestled (4 g + 1 , ℓ )-action must be spherical,and by Proposition 7.5, ℓ = 2. Therefore, Equation 4.1 in the proof ofTheorem 5.2 simplifies to give2 g + 24 g + 1 = 1 x + 1 x . Without loss of generality, we may assume that x ≤ x . Since x i | g + 1, x i ≥
3. If x = 3, then x = 3(4 g + 1)2 g + 5 = 3 (cid:18) − g + 5 (cid:19) . Since x = 3 is the only integer solution for x , Proposition 4.2 would implythat n = 3 which contradicts that fact that n = 4 g + 1. If x ≥
4, then wewould have that 12 < g g + 1 = 1 x + 1 x ≤ , which is not possible. (cid:3) Proposition 8.3.
Let F = F C F be a closed oriented surface of genus g ≥ . Let ( D , D ) be a data set pair corresponding to a root of t C ofdegree n , and let n i be the degree of D i for i = 1 , . Then the n i cannot bothsatisfy n i ≡ .Proof. Suppose for contradiction that both n i satisfy n i ≡ a i denote the a -value of D i , and let k i denote the inverse of a i modulo n i . Sincegcd( k i , n i ) = 1, the k i must be odd. Also the fact that gcd( n , n ) = 2 k forsome odd integer k implies that nn i is odd. From Equation 5.1 for the dataset pair ( D , D ), we must have that nn k + nn k ≡ n , which is impossible since nn k + nn k and n are even. (cid:3) Proposition 8.4.
Let F = F C F be a closed oriented surface of genus g ≥ . Suppose that M ( g , g ) denotes the maximum degree of a root of theDehn twist t C about C . Then M ( g , g ) ≤ g g + 4(2 g − g ) − .Proof. Let n be the order of a root of t C , given by a data set pair ( D , D ).We have n = lcm( n , n ), where n i is the degree of D i . By Remark 8.1, each n i ≤ g i +2. By Proposition 8.2, neither n i = 4 g i +1, and by Proposition 8.3,we cannot have both n = 4 g + 2 and n = 4 g + 2. If both n = 4 g and n = 4 g , then lcm( n , n ) = 4 lcm( g , g ) ≤ g g ≤ g g +4(2 g − g ) − g ≥ g , we have that M ( g , g ) ≤ max { (4 g + 2)(4 g − , (4 g − g + 2) } = 16 g g + 4(2 g − g ) − (cid:3) Notation 8.5.
We will denote the upper bound 16 g g + 4(2 g − g ) − U ( g , g ). Theorem 8.6.
Let F = F C F be a closed oriented surface of genus g ≥ .Suppose that n denotes the degree of a root of the Dehn twist t C about C .Then n ≤ g + 2 g .Proof. Since g = g − g , we have that 16 g g + 4(2 g − g ) − − g + g (16 g + 12) − (4 g + 2), which has its maximum when g = (4 g + 3). Thefact that g is an integer implies that when g is even, g = g = g/
2, andwhen g is odd, g = ( g + 1) / g = ( g − /
2. So Proposition 8.4 tells usthat when g is even, n ≤ M ( g/ , g/ ≤ g + 2 g −
2, and when g is even, n ≤ M (( g + 1) / , ( g − / ≤ g + 2 g . (cid:3) Notation 8.7.
We will denote the upper bound 4 g + 2 g derived in Theo-rem 8.6 by U ( g ).For 2 ≤ g ≤
35, Table 1 gives the realizable maximum degrees of root, m ( g ) (coming from compatible pairs of spherical nestled ( n, U ( g ). The last column gives the ratio m ( g ) /U ( g ). Thesecomputations were made using software [10] written for the GAP program-ming language. Lemma 8.8.
Suppose that we have a spherical nestled (4 g − N, -action ona F of genus g , where N is a positive odd integer. Then g ≤ N + 3 .Proof. Let D = (4 g − N, , a ; ( c , x ) , ( c , x )) be a data set for the nestled(4 g − N, F . Since 4 g − N is odd and x i | n , we have that x i ≥ x ≥
3, then Remark 4.2 implies that x ≥ (4 g − N ). So Equation 4.2gives the inequality 2 g − N + 14 g − N ≤
13 + 34 g − N , which upon simplification gives g ≤ N + 3. (cid:3) Theorem 8.9.
Let F = F C F be a closed oriented surface of genus g ≥ . Suppose that M ( g , g ) denotes the maximum order of a root of theDehn twist t C about C . Then given a positive odd integer N , we have that M ( g , g ) ≤ g g + 4(2 g − N g ) − N whenever both g i > N + 3 .Proof. By Remark 8.1, each n i ≤ g i + 2. From Propositions 8.2 and 8.3,we know that n i = 4 g i + 1 and that n i cannot both be 4 g i + 2. Suppose thatthe n i are not both even. If ℓ i >
2, then from Corollary 7.6 we have that n i ≤ (2 g i − ℓ i = 2, then Lemma 8.8 tells us that for all g i > N + 3,there exists no spherical nestled (4 g i − N, F . In particular, if g i > N + 3, then from Proposition 7.3, n i ≤ (2 g i − ≤ (2 g i − ℓ , if g i > N +3, then n i ≤ (2 g i − (2 g i − ≤ g i − N whenever g i ≥ (17 N − g i > max { N + 3 , (17 N − } = N +3, then we have that M ( g , g ) ≤ max { (4 g − N )(4 g +2) , (4 g +2)(4 g − OOTS OF DEHN TWISTS ABOUT SEPARATING CURVES 19 g m ( g ) U ( g ) m ( g ) /U ( g )2 12 20 0.603 30 42 0.714 42 72 0.585 90 110 0.816 126 156 0.817 210 210 1.008 240 272 0.889 330 342 0.9610 390 420 0.9311 462 506 0.9112 546 600 0.9113 570 702 0.8114 714 812 0.8815 798 930 0.8616 858 1056 0.8117 966 1190 0.8118 1122 1332 0.8419 1254 1482 0.8520 1326 1640 0.8121 1518 1806 0.8422 1650 1980 0.8323 1794 2162 0.8324 1950 2352 0.8325 2046 2550 0.8026 2262 2756 0.8227 2418 2970 0.8128 2550 3192 0.8029 2730 3422 0.8030 2958 3660 0.8131 3162 3906 0.8132 3306 4160 0.7933 3570 4422 0.8134 3774 4692 0.8035 3990 4970 0.80 Table 1.
The data seems to indicate that for large generathe ratio m ( g ) /U ( g ) stabilizes to the 0.79-0.82 range. N ) } = 16 g g + 4 max { (2 g − N g ) , (2 g − N g ) } − N = 16 g g + 4(2 g − N g ) − N .Suppose that both the n i are even. Then from Propositions 8.2 and 8.3,we have that M ( g , g ) ≤ lcm(4 g + 2 , g ) ≤ g g + 4 g . We need to g ( g , g m ( g , g ) U ( g , g , U ( g , g )30 (15 ,
15) 2790 3038 365831 (16 ,
15) 3162 3286 390632 (16 ,
16) 3264 3498 415832 (17 ,
15) 3162 3534 415433 (17 ,
16) 3570 3762 442233 (18 ,
15) 3534 3782 440234 (17 ,
17) 3570 3990 469034 (18 ,
16) 3774 4026 468634 (19 ,
15) 3534 4030 465035 (18 ,
17) 3990 4270 497035 (19 ,
16) 3876 4290 495035 (20 ,
15) 3690 4278 4898
Table 2.
For N = 11, this data illustrates the stable bound U ( g , g ,
11) and the upper bound U ( g , g ). When g = 32,we saw in Table 1 that the maximum realizable degree m ( g ) = 3306. This is larger than both the stable bounds U (16 , ,
11) and U (17 , , g g + 4 g ≤ g g + 4(2 g − N g ) − N . Since g > N + 3,(16 g g +4(2 g − N g ) − N ) − (8 g g +4 g ) = 8 g g +8 g − N +1) g − N > g g + 8 g + 4( g − g + 2( g −
3) = 12 g g + 10 g − g − > (cid:3) Notation 8.10.
We will denote the upper bound 16 g g +4(2 g − N g ) − N derived in Theorem 8.9 by U ( g , g , N ). Example 8.11.
When N = 11, if both g i >
14, then from Theorem 8.9, M ( g , g ) ≤ U ( g , g ,
11) = 16 g g + 4(2 g − g ) −
22. For genera pairs( g , g ) with 30 ≤ g + g ≤
35, Table 2 gives the values of the realizablemaximum degree m ( g , g ) (coming from compatible spherical nestled ( n, U ( g , g ) (derived in Proposition 8.4), and thestable upper bound U ( g , g , N ). acknowledgements I would like to thank Steven Spallone for some useful discussions in ele-mentary number theory.
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Department of Mathematics, University of Oklahoma, Norman, Oklahoma73019, USA
URL : e kashyap/ E-mail address ::