FACTORING IN THE HYPERELLIPTIC TORELLI GROUP

Transcription

1 FACTORING IN THE HYPERELLIPTIC TORELLI GROUP TARA E. BRENDLE AND DAN MARGALIT Abstract. The hyperelliptic Torelli group is the subgroup of the mapping class group consisting of elements that act trivially on the homology of the surface an that also commute with some fixe hyperelliptic involution. The authors an Putman prove that this group is generate by Dehn twists about separating curves fixe by the hyperelliptic involution. In this paper, we introuce an algorithmic approach to factoring a wie class of elements of the hyperelliptic Torelli group into such Dehn twists, an apply our methos to several basic elements. 1. Introuction Let s : Sg 1 Sg 1 be a hyperelliptic involution of a surface of genus g with one bounary component; see Figure 1. The hyperelliptic Torelli group SI(Sg 1 ) is the groupofhomeomorphismsofsg 1 thatcommutewiths,restricttotheientityon Sg, 1 an act trivially on H 1 (Sg 1 ), moulo isotopy. This group arises as the funamental group of each component of the branch locus of the perio mapping an also as the kernel of the Burau representation at t = 1; see [7].... Figure 1. Rotation by π about the inicate axis is a hyperelliptic involution. The simplest nontrivial element of SI(Sg 1 ) is a Dehn twist about a symmetric separating curve, that is, a separating curve fixe by s. Hain conjecture SI(Sg) 1 is generate by such elements, an the authors recently prove this conjecture with Putman [5]. There are two other basic elements of SI(Sg): 1 Symmetric simply intersecting pair maps: If x an y are symmetric nonseparating curves with vanishing algebraic intersection î(x, y), then the commutator of their Dehn twists [T x,t y ] lies in SI(Sg 1 ); see Figure 2. Symmetrize simply intersecting pair maps: If u 1, v 1, u 2, an v 2 are nonseparating curves with u 1 v 1 = 2, î(u 1,v 1 ) = 0, s(u 1 ) = u 2, s(v 1 ) = v 2, an(u 1 v 1 ) (u 2 v 2 ) =, then[t u1 T u2,t v1 T v2 ] lies in SI(Sg 1 ); seefigure Mathematics Subject Classification. Primary: 20F36; Seconary: 57M07. Key wors an phrases. Torelli group, hyperelliptic involution, Dehn twists. The first author is partially supporte by the EPSRC. The secon author is supporte by the Sloan Founation an the National Science Founation. 1

2 2 TARA E. BRENDLE AND DAN MARGALIT When the authors first learne of Hain s conjecture, it seeme intractable because we i not know how to factor these elements into Hain s propose generators. In this paper, we not only give relatively simple factorizations for both, but we also give an algorithm for factoring a much wier class of elements. We expect that our relations will play a role for SI(S 1 g) analogous to the critical role that the classical lantern relation has playe in our unerstaning of the full Torelli group. The group SI(S 1 g) is isomorphic to the kernel of the Burau representation of the brai group B 2g+1 evaluate at t = 1. In another paper [2], we use the iea from our factorization algorithm to erive a square lantern relation an we use it to characterize the kernel of the Burau representation at t = 1, moulo 4. Higher genus twists. The genus of a separating curve in Sg 1 is the genus of the complementary component not containing Sg. 1 In our earlier paper [4], we showe that a Dehn twist about a symmetric separating curve of arbitrary genus is equal to a prouct of Dehn twists about symmetric separating curves of genus 1 an 2. In particular, by our theorem with Putman, SI(Sg 1 ) is generate by Dehn twists about such curves. As an application of the methos of this paper, we give an explicit factorization of the Dehn twist about any genus k 3 symmetric separating curve into Dehn twists about symmetric separating curves of smaller genus. x u 2 v2 y u 1 v 1 Figure 2. Left: The curves x an y form a symmetric simply intersecting pair. Right: the curves u 1, v 1, u 2, an v 2 form a symmetrize simply intersecting pair. Algorithmic factorizations. Let abeasymmetricnonseparatingcurveinsg 1, an enote by SI(Sg,a) 1 the stabilizer of the isotopy class of a in SI(Sg). 1 There is an s-equivariant inclusion Sg 1 a S1 g 1 an this inuces a surjective homomorphism SI(Sg,a) 1 SI(Sg 1 1 ) [4, Proposition 6.6]. We enote the kernel by SIBK(S1 g,a): 1 SIBK(S 1 g,a) SI(S1 g,a) SI(S1 g 1 ) 1. Theorem 1.1. There is an explicit algorithm for factoring arbitrary elements of SI(S 1 g,a) into Dehn twists about symmetric separating curves of genus 1 an 2. The iea is to ientify SIBK(Sg 1,a) with a subgroup of the funamental group of a isk with 2g 1 points remove. Then the problem of factoring elements of SIBK(Sg 1,a) into Dehn twists about symmetric separating curves is translate into a problem of fining special factorizations of certain elements of this free group. Given any symmetric simply intersecting pair map or symmetrize simply intersecting pair map, we can fin a curve a so that the given map lies in SIBK(Sg 1,a); choose a to be the core of any annular region in the complement of the union of the

3 FACTORING IN THE HYPERELLIPTIC TORELLI GROUP 3 efining curves of the map. Therefore, we can unerstan both types of maps in the context of Theorem 1.1. Ifcisagenusk symmetricseparatingcurveinsg, 1 thenwecanchooseagenusk 1 symmetric separating curve an a symmetric nonseparating curve a so that T c T 1 lies in the corresponing SIBK(Sg 1,a); we take a to lie in the genus 1 subsurface between c an. Therefore, by Theorem 1.1, we can factor T c T 1 into a prouct of Dehn twists about symmetric separating curves of genus 1 an 2. We emphasize that the existence of this algorithm oes not guarantee that one can fin a simple factorization for a given element of SIBK(Sg,a). 1 The factorizations we give in this paper were only foun after much trial an error (cf. [3]). Their relative tameness suggests that SI(Sg 1 ) is more tractable than originally believe. Finally, we can obtain factorizations in the hyperelliptic Torelli group of a close surface S h by incluing Sg 1 into S h where h g; the inuce map SI(Sg 1) SI(S h) is injective if h > g an has cyclic kernel T S 1 g otherwise; see [4, Theorem 4.2]. Acknowlegments. First, we woul like to thank Anrew Putman for suggesting the problem of factoring the symmetrize simply intersecting pair maps. We woul also like to thank Mlaen Bestvina, Joan Birman, Leah Chilers, an an anonymous referee for helpful comments an conversations. Our work was greatly aie by the Maple program β-twister, written by Marta Aguilera an Juan González-Meneses. 2. The factoring algorithm In this section we explain how to algorithmically factor an arbitrary element of SIBK(Sg 1,a) as per Theorem 1.1. The setup. We woul like to rephrase our problem about factoring elements of SI(Sg,a) 1 into a problem about certain factorizations in a free group. We start by giving some efinitions an then outlining the iea. Let D 2g 1 enote a isk with 2g 1 marke points an D2g 1 the puncture isk obtaine by removing the marke points. The funamental group of D2g 1 is a free group F 2g 1 ; we take the generators x 1,...,x 2g 1 for F 2g 1 to be simple loops in D 2g 1 each surrouning one marke point; see Figure 3. x 2g 1 x 2g 2 x 1 p Figure 3. The generators x i for π 1 (D 2g 1 ). Let F2g 1 even enote the kernel of the map F 2g 1 Z/2Z given by x i 1 for all i; this group is generate by the x δ i i xδ j j with i j an δ i { 1,1}. Denote the

4 4 TARA E. BRENDLE AND DAN MARGALIT generators for Z 2g 1 by e 1,...,e 2g 1. Let ǫ : F2g 1 even Z2g 1 be the homomorphism given by x δ i i xδ j j e i e j. We will require the following two facts, explaine below. (1) There is an isomorphism Ψ : SIBK(Sg 1,a) kerǫ. (2) kerǫ is generate by squares of simple loops in D2g 1 about 1 or 3 punctures. Once we efine Ψ, it will be easy to see that squares of simple loops in D 2g 1 surrouning 1 or 3 punctures correspon to proucts of Dehn twists about symmetric separating curves in S 1 g of genus 1 an 2. After iscussing the above two facts, we procee to explain the factorization algorithm of Theorem 1.1. The isomorphism Ψ. The isomorphism Ψ was given in our earlier paper [4, Theorem 1.2]; we recall the construction. In what follows, the mapping class group of a surface S is the group Mo(S) of isotopy classes of homeomorphisms of S that restrict to the ientity on S an preserve the set of marke points. The quotient Sg/ s 1 is a isk D 2g+1 with 2g+1 marke points, an Mo(D 2g+1 ) is isomorphic to the brai group B 2g+1. Let SMo(Sg 1) be the subgroup of Mo(S1 g ) with elements represente by s-equivariant homeomorphisms. Birman Hilen prove the natural map Θ : SMo(Sg 1) B 2g+1 is an isomorphism [6, Theorem 9.1]. The group SI(Sg,a) 1 maps to Mo(D 2g+1,a), the stabilizer of the isotopy class of the arc a, the image of a in D 2g+1. By collapsing a to a marke point p an removing the other 2g 1 marke points to obtain 2g 1 punctures, we obtain a homomorphism Ξ : Mo(D 2g+1,a) Mo(D2g 1,p). Since the kernel of Ξ Θ is generate by T a, the restriction Ψ : SI(Sg,a) 1 Mo(D2g 1,p) is injective. We then arrive at the following special case of the Birman exact sequence: 1 π 1 (D 2g 1,p) Mo(D 2g 1,p) Mo(D 2g 1 ) 1. The first nontrivial map here is actually an anti-homomorphism, as the usual orers of operation in the two groups o not agree. Therefore, relations in π 1 (D2g 1,p) will translate to the reverse relations in Mo(D2g 1,p). The image of SIBK(Sg,a) 1 uner Ψ lies in the kernel π 1 (D2g 1,p) of the Birman exact sequence. In our earlier paper [4, Lemma 4.5] we showe that for α π 1 (D2g 1,p), the action of the lift (Ξ Θ) 1 (α) on H 1 (Sg 1 ;Z) is exactly given by ǫ(α), an so the image of Ψ is precisely kerǫ. Squares of simple loops. If α π 1 (D2g 1,p) is a simple loop surrouning k punctures, where k is o, then α 2 lies in kerǫ = ImΨ an Ψ 1 (α 2 ) is equal to T c T 1, where c an are the preimages in S1 g of the curves obtaine by pushing α off of p to the left an right, respectively, an positive Dehn twists are to the left. The curves c an are separating curves of genus (k+1)/2 an (k 1)/2 (not necessarily in that orer) an a lies in the genus 1 subsurface between them. When k = 1, note that one of the two separating curves is inessential. We will now show that these α 2 generate kerǫ. To begin, the image of ǫ is Z 2g 1 bal, the kernel of the map Z 2g 1 Z recoring the coorinate sum [4, Lemma 5.1].

5 FACTORING IN THE HYPERELLIPTIC TORELLI GROUP 5 Also, the group F even 2g 1 is generate by elements of the form x2 i an the x jx 1, since an Z 2g 1 bal x i x j = (x i x 1 )(x j x 1 ) 1 (x 2 j ) an x ix 1 j = (x i x 1 )(x j x 1 ) 1, has a presentation whose generators are the images of these generators: ǫ(x 2 1 ),...,ǫ(x2 2g 1 ),ǫ(x 2x 1 ),...,ǫ(x 2g 1 x 1 ) ǫ(x 2 i ),[ǫ(x ix 1 ),ǫ(x j x 1 )]. It follows that kerǫ is normally generate by the set {x 2 i 1 i 2g 1} {[x ix 1,x j x 1 ] 1 i < j 2g 1}. We notice the following relation in kerǫ: [x i x 1,x j x 1 ] = [x 2 j (x j x i x 1 ) 2 (x 2 i ) x 1 1 x 2 1 ]x j, where x y enotes yxy 1. It now follows that kerǫ is normally generate by {x 2 i 1 i 2g 1} {(x jx i x 1 ) 2 1 i < j 2g 1}. Referring to Figure 3, we see that each x j x i x 1 is a simple close curve in D 2g 1 when j > i. In particular, kerǫ is generate by {α 2 α is a simple loop surrouning 1 or 3 punctures}. It follows that SIBK(Sg,a) 1 is generate by maps of the form T c where c is a symmetric separating curve of genus 1 with a lying on the genus 1 sie of c an of the form T c T 1 where c an are symmetric separating curves of genus 1 an 2, respectively, with a lying in the genus 1 subsurface between. The algorithm. We now give an algorithm for factoring arbitrary elements of ker ǫ in terms of squares of simple loops in D2g 1, each surrouning 1 or 3 punctures. Suppose we are given some f SIBK(Sg,a) 1 as a prouct of Dehn twists about symmetriccurvesinsg. 1 Wecanrealizef asanelementf ofthegroupmo(d2g 1,p) using the following ictionary: a Dehn twist about a symmetric nonseparating curve c correspons to a half-twist about the image arc c in D2g 1 an a Dehn twist about a symmetric separating curve correspons to the square of the Dehn twist about the image curve in D2g 1. As above, since f SIBK(S1 g,a), we know that f lies in the kernel of the above Birman exact sequence. We can then use Artin s combing algorithm for pure brais [1] to write f as a wor w 0 in the x ±1 i. As above, the wor w 0 lies in F2g 1 even, an so it equals some wor w in the x2 i an the x i x 1. Since Ψ(f) kerǫ, the wor w maps to a relator in Z 2g 1 bal with respect to the presentation given above, that is, w maps to a wor in the generators ǫ(x 2 i ) an ǫ(x i x 1 ) that equals the ientity. Therefore, there is a sequence of commutations, free cancellations, an cancellations of ǫ(x 2 i )-terms transformingǫ(w) into the empty wor (this is the obvious solution to the wor problem for a free abelian group). By the corresponence between relators for Z 2g 1 bal an normal generators for ker ǫ, we obtain a factorization of w into a prouct of conjugates of the x 2 i, the [x ix 1,x j x 1 ], an their inverses. We alreay explaine above how to factor [x i x 1,x j x 1 ] into a prouct of squares of simple loops surrouning 1 or 3 punctures, so we are one.

6 6 TARA E. BRENDLE AND DAN MARGALIT Relations in genus two. Above, we explaine how the commutator [x i x 1,x j x 1 ] correspons to an element of SI(S2 1 ) an we factore this into a prouct of five Dehn twists about symmetric separating curves in S2 1 (see [3, Theorem 3.1] for a picture of the curves). If we cap the bounary of S2 1 with a isk, [x ix 1,x j x 1 ] maps to [Tc 2T 2 e,tb 2T 2 ]T8 a in SI(S 2); see Figure 4. The relate (but simpler) element [T c Te 1,T b T 1 ]T2 a also lies in SI(S 2) an so is a prouct of Dehn twists about symmetric separating curves. Surprisingly, it equals a single (left) Dehn twist. c b a e f Figure 4. The curves a, b, c,, e, an f from Theorem 2.1. Theorem 2.1. Let a, b, c,, e, an f be as in Figure 4. We have: [T c T 1 e,t b T 1 ]T2 a = T f. One can check the relation in Theorem 2.1 using the Alexaner Metho[6, Section 2.3]; see [3, Section 4.1] for a conceptual proof. 3. Applications In this section, we give explicit factorizations of symmetric simply intersecting pair maps an symmetrize simply intersecting pair maps into Dehn twists about symmetric separating curves. We also give an explicit factorization of the Dehn twist about any genus k 3 symmetric separating curve into Dehn twists about symmetric separating curves of smaller genus Factoring symmetric simply intersecting pair maps. We start by writing the symmetric simply intersecting pair map from Figure 5 as a prouct of Dehn twists about symmetric separating curves. x w y a v Figure 5. The curves use in Theorem 3.1. Theorem 3.1. Every symmetric simply intersecting pair map is the prouct of two Dehn twists about symmetric simple close curves. In particular, if x, y, v, an w are the simple close curves shown in Figure 5, we have: [T x,t y ] = T 1 v T w.

7 FACTORING IN THE HYPERELLIPTIC TORELLI GROUP 7 Note that the first statement follows immeiately from the secon statement an the change of coorinates principle [6, Section 1.3]. We give two proofs of Theorem 3.1. The first is an easy application of the lantern relation, a relation between (left) Dehn twists about 7 curves lying in a subsurface homeomorphic to a sphere with four bounary components; see [6, Section 5.1.1]. First proof of Theorem 3.1. By the lantern relation, we have T v T x T y = M an T w T y T x = M, where M an M are the proucts of twists about the bounary curves of the corresponing four-hole spheres. Since x an y appear in both lantern relations an since a regular neighborhoo of x y is a sphere with four holes, the four-hole spheres in the two lantern relations are equal, an so M = M. Thus, as esire. [T x,t y ] = (T x T y )(T 1 x T 1 y ) = (T 1 v M)(M 1 T w ) = T 1 v T w, We now give a proof of Theorem 3.1 that is intrinsic to the brai group. y δ q 1 q 2 x γ w α β v e Figure 6. Curves, loops, an arcs use in the secon proof of Theorem 3.1 Secon proof of Theorem 3.1. The images of x an y in D 2g+1 are arcs x an y; enote their enpoints by q 1 an q 2 (throughout, refer to Figure 6). As above, T x an T y correspon to the half-twists H x an H y in Mo(D 2g+1 ). As a loop in the space of configurations of 2g+1 points in the isk (see [6, Theorem 9.1]), the prouct H x H y is given by the motion of points where q 1 an q 2 move aroun δ an γ, respectively (we multiply half-twists right to left). These motions correspon to the mapping classes Tv 1 T an Tv 1 T e, respectively. Similarly, Hx 1 H 1 y correspons to (T w T 1 )(T wte 1 ). Since T an T e commute with all the other twists, the original commutator [T x,t y ] in SI(Sg) 1 correspons to Tv 2 Tw 2 in Mo(D 2g+1 ). The preimage uner Ψ is Tv 1 T w in SMo(Sg 1 ), as esire. The secon proof of Theorem 3.1 has a connection with the algorithm from Section 2. Let a be the symmetric simple close curve in Sg 1 shown in Figure 5. Assuming Theorem 3.1, an referring to Figure 6, we can see that Tv 2 Tw 2, the image of Tv 1 T w uner Ψ, is α 2 β 2, where α an β are as shown in Figure 6. This is a prouct of squares of simple loops, each surrouning one puncture, as per Section 2.

8 8 TARA E. BRENDLE AND DAN MARGALIT 3.2. Factoring symmetrize simply intersecting pair maps. We now aress the symmetrize simply intersecting pair maps. Theorem 3.2. Every symmetrize simply intersecting pair map is equal to a prouct of six Dehn twists about symmetric separating curves. u 2 v2 v u u 1 v 1 a e β δ α γ Figure 7. The curves an loops use in the proof of Theorem 3.2. Proof. Consier the symmetrize simply intersecting pair map shown in Figure 7 (throughout we refer to this figure). First we notice that [T u1 T u2,t v1 T v2 ] lies in SIBK(Sg,a). 1 We claim that the image of this commutator uner the map Ψ from Section 2 is [δ,γ]. Inee, we have Ψ(T u1 T u2 ) = T u an Ψ(T v1 T v2 ) = T v, an so the claim follows from the fact that the images of δ an γ in Mo(D2g 1,p) are Tv 1 T v an Tu 1 T u an the fact that T u an T v commute with all other twists in the commutator (remember that the orer of multiplication gets reverse!). Now that we have written Ψ([T u1 T u2,t v1 T v2 ]) as an element of the free group π 1 (D2g 1,p), we observe the following factorization in this free group: [δ,γ] = [β 2 (β 1 γ) 2 (αγ) 2 α 2 ] α As in Section 2, this is a prouct of squares of simple loops in π 1 (D2g 1,p) surroning 1 or 3 punctures, an hence the preimage uner Ψ is a prouct of Dehn twists about symmetric separating curves of genus 1 an 2 in Sg. 1 The first an fourth loops each correspon to a single Dehn twist, while the secon an thir loops each correspon to a pair of Dehn twists. From the proof of Theorem 3.2, it is straightforwar, though not necessarily enlightening, to raw the six symmetric separating simple close curves whose Dehn twists factorize the symmetrize simply intersecting pair map shown in Figure 2. For an explicit picture, see the first version of this paper [3]. In terms of the x i from Figure 3, we can also write the Ψ-image of a symmetrize simply intersecting pair map as [x 4 x 3,x 2 x 1 ]. In the free group, this factors as: [x 4 x 3,x 2 x 1 ] = [(x 3 x 2 x 1 ) 2 (x 2 3 )(x 2x 1 ) 1 (x 4 x 2 x 1 ) 2 (x 2 4)] x 4.

9 FACTORING IN THE HYPERELLIPTIC TORELLI GROUP 9 Again, the right-han sie in this equality is a prouct of squares of simple loops an so we obtain an alternate factorization. The relation given in Theorem 3.2 involves 14 Dehn twists, twice the number of Dehn twists involve in the lantern relation. However, there is no way to rearrange our relation into a prouct of two lantern relations Factoring higher genus twists. Finally, we obtain the factorization of a Dehn twist about an arbitrary symmetric separating curve into a prouct of Dehn twists about symmetric separating curves, each having genus 1 or 2, by applying the following theorem inuctively. Theorem 3.3. Let enote the bounary of Sg 1, an let c enote a symmetric separating curve of genus g 1. The prouct T Tc 1 is equal to a prouct of 10 Dehn twists about symmetric separating curves in Sg, 1 each of genus at most g 1. Proof. Let a be a symmetric nonseparating curve in Sg 1 lying between c an. The image of in D 2g+1 is the bounary of the isk, an if we choose the ientification of Sg/ s 1 with D 2g+1 appropriately, the image of c in D 2g+1 is a roun circle surrouning the 2g 1 leftmost marke points an the image of a is a straight arc connecting the other two marke points. Let y 5 enote x 2g 1 x 2g x 5. It follows from the previous paragraph that the image of T Tc 1 uner Ψ is equal to the image of (y 5 x 4 x 3 x 2 x 1 ) 2 uner the point pushing map π 1 (D2g 1,p) Mo(D 2g 1,p). As in Section 2, we factor (y 5 x 4 x 3 x 2 x 1 ) 2 into a prouct of simple loops each surrouning an o number of punctures: [(x 2 x 1 y 5 ) 2 (x 2 2 )y 1 5 x 1 1 (x3 x 1 y 5 ) 2 x 2 3] y 5x 4 x 3 [(x 1 y 5 x 4 ) 2 (x 2 4 )(x 4x 3 x 2 ) 2 ] x 1 1. This is a prouct of 11 Dehn twists about symmetric separating curves, three of genus g 1, three of genus g 2, four of genus 1, an one of genus 2. References [1] Emil Artin. Theory of brais. Ann. of Math. (2), 48: [2] Tara Brenle an Dan Margalit. Congruence subgroups of brai groups. Preprint. [3] Tara Brenle an Dan Margalit. Factoring in the hyperelliptic Torelli group. arxiv: v1. [4] Tara Brenle an Dan Margalit. Point pushing, homology, an the hyperelliptic involution. to appear in Michigan Mathematical Journal. [5] Tara Brenle, Dan Margalit, an Anrew Putman. Generators for the hyperelliptic Torelli group an the kernel of the Burau representation at t = 1. Preprint. [6] Benson Farb an Dan Margalit. A primer on mapping class groups. Princeton Univ. Press, [7] Richar Hain. Finiteness an Torelli spaces. In Problems on mapping class groups an relate topics, volume 74 of Proc. Sympos. Pure Math., pages Amer. Math. Soc., Provience, RI, Tara E. Brenle, School of Mathematics an Statistics, 15 University Garens, University of Glasgow, G12 8QW, tara.brenle@glasgow.ac.uk Dan Margalit, School of Mathematics, Georgia Institute of Technology, 686 Cherry St., Atlanta, GA 30332, margalit@math.gatech.eu