A positive proportion of elements of mapping class groups is pseudo-Anosov

Bull. London Math. Soc. 50 (2018) 390–394 C e2018 London Mathematical Society doi:10.1112/blms.12148 A positive proportion of elements of mapping class groups is pseudo-Anosov Mar´ıa Cumplido and Bert Wiest Abstract In the Cayley graph of the mapping class group of a closed surface, with respect to any generating set, we look at a ball of large radius centered on the identity vertex, and at the proportion among the vertices in this ball representing pseudo-Anosov elements. A well-known conjecture states that this proportion should tend to one as the radius tends to infinity. We prove that it stays bounded away from zero. We also prove similar results for a large class of subgroups of the mapping class group. 1. Pseudo-Anosovs in mapping class groups Let Gbe the mapping class group of a closed surface Σ (with genus(Σ) ⩾2), equipped with any finite generating set S. It is a widely held belief that ‘most’ elements of Gare pseudo-Anosov. In order to make this statement precise, we have to specify a way picking a ‘random element’ of G. There are at least two standard methods for doing so, both involving the Cayley graph Γ of Gwith respect to S. The first method is to perform a random walk in the Cayley graph Γ of G, starting at the identity vertex. In this framework, the belief is that the probability of obtaining a pseudo-Anosov element tends (exponentially quickly) to 1 as the length of the random walk tends to infinity. This belief has been proven to be correct, and it has been generalised far beyond the realm of mapping class groups, in [8,10,11]. For the second method, consider the ball BΓ(1,R) in Γ of radius Rand centered on the identity vertex 1. The belief now is that the proportion of pseudo-Anosov elements among the vertices in this ball tends to 1 as the radius tends to infinity: lim R→∞ |pseudo-Anosov elements of G∩BΓ(1,R)| |BΓ(1,R)|=1. This statement remains a conjecture despite some progress in [4]. Our main result is Corollary 1. lim inf R→∞ |pseudo-Anosov elements of G∩BΓ(1,R)| |BΓ(1,R)|>0. Our proof is actually very easy, and it uses only well-known ingredients — notably, a classical theorem of Albert Fathi [7], and the acylindrical hyperbolicity of the mapping class group action on the curve complex [3]. Therefore it seems likely that Corollary 1,andourproof,is well known to some experts. However, we have not been able to locate it in the literature, and believe it deserves to be written down. Received 12 September 2017; published online 28 March 2018. 2010 Mathematics Subject Classification 20F65, 57S30 (primary), 57M07, 57M60, 20P05 (secondary). Mar´ıa Cumplido was supported by a PhD contract founded by Universit´e Rennes 1, Spanish Projects MTM2013-44233-P, MTM2016-76453-C2-1-P and FEDER and the French-Spanish mobility programme “M´erim´ee 2015”. 14692120, 2018, 3, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/blms.12148 by Universidad De Sevilla, Wiley Online Library on [08/11/2022]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License A POSITIVE PROPORTION OF ELEMENTS OF MAPPING CLASS GROUPS 391 We will deduce Corollary 1from the following, which is our main technical result: Theorem 2 (Positive density of pseudo-Anosovs).There exists a finite subset Fof Gsuch that for any element gof G, at least one of the mapping classes in {f◦g|f∈F}is pseudo- Anosov. The proofs of Corollary 1and Theorem 2will be given in Section 2. In Section 3we will show that analogue results also hold in a large class of subgroups of mapping class groups, notably in finite index subgroups of G, and in the Torelli group. 2. Proofs ProofofTheorem2. In what follows, by a curve we always mean an isotopy class of simple closed curves. We start by recalling the result from [7]: Theorem 3 (Fathi).If g∈G and if cis a simple closed curve on Σsuch that the curves {gn(c)|n∈Z}together fill Σ, then Tk c◦gis always pseudo-Anosov, except for at most seven consecutive values of k(where Tcdenotes the Dehn twist along the curve c). In particular, either gor T7 c◦g(or both)are pseudo-Anosov. Lemma 4. There are two curves a, b in Σsuch that for all but finitely many elements gof G, either aand g(a)together fill Σ,orband g(b)together fill Σ(or both). Proof of Lemma 4. Recall that two curves together fill Σ if and only if, in the curve complex CC(Σ), they are at distance at least 3. Next we recall a fundamental result due to Bowditch [3]: the mapping class group Gacts acylindrically on the curve complex, that is, for any r⩾0, there exist R(r),N(r)⩾0 so that for any two vertices a,bof the curve complex with dCC(a, b)>R(r) there are at most N(r) distinct elements gof Gsuch that dCC(a, g(a)) <rand dCC(b, g(b)) <r. We will apply this result in the case r= 3. We simply choose arbitrarily two curves aand bin Σ, the only restriction being that dCC(a, b)>R(3). Bowditch’s result says that for all but finitely many (at most N(3)) elements gof G, the action of gon the curve complex displaces least one of the two vertices, aor b, by at least 3. This completes the proof of Lemma 4. Now we define F={1G∪T7 a∪T7 b}. Fathi’s theorem and Lemma 4together imply the following: for every element gof G, apart from the finitely many exceptional ones, at least one of the mapping classes in {f◦g|f∈F }is pseudo-Anosov. Now we have to deal with the finitely many exceptional elements of G,say{g1,...,g N}, where N⩽N(3). For each gi, we choose one fi∈G such that fi◦giis not an exceptional element. Then for every i, at least one of the mapping classes in {f◦fi◦gi|f∈F }is pseudo-Anosov. Thus we have completed the proof of Theorem 2, with F=1,T7 a,T7 b i=1,...,N fi,T 7 a◦fi,T 7 b◦fi. Proof of Corollary 1. Let R= max {dΓ(1,f)|f∈F}+ 1. Theorem 2tells us that the union of all the balls of radius Rcentered on the pseudo-Anosov vertices of Γ is the whole Cayley graph Γ. Now for R>R , the centers of the balls of radius Rcovering BΓ(1,R−R) have to lie in BΓ(1,R). Thus the union of the balls of radius Raround the pseudo-Anosov vertices of 14692120, 2018, 3, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/blms.12148 by Universidad De Sevilla, Wiley Online Library on [08/11/2022]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 392 MAR´ IA CUMPLIDO AND BERT WIEST BΓ(1,R) contains BΓ(1,R−R). We obtain |BΓ(1,R )|·|pseudo-Anosov elements of G∩BΓ(1,R)|⩾|BΓ(1,R−R)| and hence |pseudo-Anosov elements of G∩BΓ(1,R)| |BΓ(1,R)|⩾|BΓ(1,R−R)| |BΓ(1,R)|·1 |BΓ(1,R )|. Both factors can be bounded below independently of R. Indeed, since all vertices of Γ are of valence at most 2|S|, we have the very rough estimate ⩾1 2|S|+(2|S|)2+···+(2|S|)R·1 1+2|S|+(2|S|)2+···+(2|S|)R⩾1 (2|S|)2(R+1) . Thus the lim inf studied in Corollary 1is at least (2 |S|)−2(R+1). Remark 1. After a first version of this paper appeared as a preprint, Mladen Bestvina kindly pointed out to us that there is a completely different proof of Theorem 2.Thisproof has the virtue of also applying to other contexts, notably to proving that fully irreducible elements have positive density in Out(Fn). The key is the existence [2] of homogeneous quasimorphisms ϕ:G→Rwhich are unbounded, but so that |ϕ|is uniformly bounded by some constant Con all reducible and periodic elements. Now simply fix such a quasi-morphism ϕ, and some element fof Gwith ϕ(f)>C+Δ and ϕ(f−1)<−C−Δ (where Δ denotes the defect of ϕ). Then for any g∈G, either f−1◦gor f◦gis pseudo-Anosov. For details in the Out(Fn)-context see [1], particularly Remark 4.33. Remark 2. If we want to adapt our method of proof to the Out(Fn)-context, we need an analogue of Fathi’s theorem in this setting. Unfortunately, no such result seems to be known. Note that [5, Proposition 3.1] is insufficient for this purpose, as it yields no universal bound on the number of exceptional twists. 3. Pseudo-Anosovs in subgroups of mapping class groups Throughout this section, we take Hto be a subgroup of G, and we take Γ Hto be the Cayley graph of Hwith respect to any fixed finite generating set. We will be interested in the case when Hsatisfies the following condition: Condition (∗) There are vertices a, b of the curve complex of CC(Σ) and integers ka,k b∈Z such that •dCC(a, b)⩾R(3) and •Tka a,Tkb b∈H were R(3) is the number used in the proof of Lemma 4in Section 2. Example 1. Condition (∗) is for instance satisfied •if His a finite index subgroup of G,or •if His the Torelli group — see [6, Chapter 6]. Note that the Torelli group contains in particular all Dehn twists along separating curves of Σ, or •if His any finite index subgroup of the Torelli group. Theorem 5. Suppose His a subgroup of G, the mapping class group of Σ, satisfying Condition (∗)above. Then pseudo-Anosovs have positive density in H: there exists a finite 14692120, 2018, 3, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/blms.12148 by Universidad De Sevilla, Wiley Online Library on [08/11/2022]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License A POSITIVE PROPORTION OF ELEMENTS OF MAPPING CLASS GROUPS 393 subset Fof Hsuch that for any element gof H, at least one of the mapping classes in {f◦g|f∈F}is pseudo-Anosov. Moreover, lim inf R→∞ |pseudo-Anosov elements of H∩BΓH(1,R)| |BΓH(1,R)|>0 Proof. The proof follows very closely the proof of Theorem 2. This time we choose the vertices aand bof Σ, as well as integers kaand kbaccording to Condition (∗). Possibly after replacing kaand kbby integer multiples of themselves, we can assume moreover that ka,k b⩾7. Defining F={1H∪Tka a∪Tkb b}, Fathi’s theorem and Lemma 4now imply: for every element gof H, apart from the finitely many exceptional ones, at least one of the mapping classes in {f◦g|f∈F }is pseudo-Anosov. For each exceptional element gi∈H,wechooseonefi∈H such that fi◦gi∈H is nonexceptional. Now with the exact same argument as in the proof of Theorem 2we have: for every g∈H, at least one of the mapping classes in {f◦g|f∈F }is pseudo-Anosov, with F=1,Tka a,Tkb b i=1,...,N fi,T ka a◦fi,T kb b◦fi. Finally, the deduction that the limit inferior of the proportion of pseudo-Anosov elements in large balls in the Cayley graph of His strictly positive works literally as the proof of Corollary 1, just replacing Gby Hand Γ by ΓHthroughout.  Remark 3. The proof of Theorem 2did not use the full strength of Bowditch’s theorem [3] — knowing that there exists one element of Gwhich acts in a weakly properly discontinuous (WPD) manner on the curve complex [9] was more than sufficient. However, for our proof of Theorem 5, Bowditch’s theorem is needed almost in its full strength. Acknowledgements. We thank Anna Lenzhen for a helpful conversation, and Mladen Bestvina for the email mentioned in Remark 1. References 1. M. Bestvina and M. Feighn, ‘A hyperbolic Out(Fn)-complex’, Groups Geom. Dyn. 4 (2010) 31–58. 2. M. Bestvina and K. Fujiwara, ‘Quasi-homomorphisms on mapping class groups’, Glas. Mat. Ser. III 42(62) (2007) 213–236. 3. B. Bowditch, ‘Tight geodesics in the curve complex’, Invent. Math. 171 (2008) 281–300. 4. S. Caruso and B. Wiest, ‘On the genericity of pseudo-Anosov braids II: conjugations to rigid braids’, Groups Geom. Dyn. 11 (2017) 549–565. 5. M. Clay and A. Pettet, ‘Current twisting and nonsingular matrices’, Comment. Math. Helv. 87 (2012) 385–407. 6. B. Farb and D. Margalit,A primer on mapping class groups, Princeton Mathematical Series 49 (Princeton University Press, Princeton, NJ, 2012). 7. A. Fathi, ‘Dehn twists and pseudo-Anosov diffeomorphisms’, Invent. Math. 87 (1987) 129–151. 8. J. Maher, ‘Exponential decay in the mapping class group’, J. Lond. Math. Soc. (2) 86 (2012) 366– 386. 9. P. Przytyicki and A. Sisto, ‘A note on acylindrical hyperbolicity of mapping class groups’, Advanced Studies in Pure Mathematics, Hyperbolic Geometry and Geometric Group Theory conference proceedings, 255–264. 10. I. Rivin, ‘Walks on groups, counting reducible matrices, polynomials, and surface and free group automorphisms’, Duke Math. J. 142 (2008) 353–379. 11. A. Sisto, ‘Contracting elements and random walks’, J. Reine Angew. Math.,https://doi.org/10.1515/ crelle-2015-0093. 14692120, 2018, 3, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/blms.12148 by Universidad De Sevilla, Wiley Online Library on [08/11/2022]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 394 MAR´ IA CUMPLIDO AND BERT WIEST Mar´ıa Cumplido Univ Rennes, CNRS IRMAR - UMR 6625 F-35000 Rennes France and Depto. de ´ Algebra Instituto de Matem´aticas (IMUS) Universidad de Sevilla Av. Reina Mercedes s/n 41012 Sevilla Spain maria.cumplidocab[email protected] Bert Wiest Univ Rennes, CNRS IRMAR - UMR 6625 F-35000 Rennes France b[email protected] 14692120, 2018, 3, Downloaded from https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/blms.12148 by Universidad De Sevilla, Wiley Online Library on [08/11/2022]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License