Genericity of contracting elements in groups

Transcription

1 Genericity of contracting elements in groups Wenyuan Yang (Peking University) 2018 workshop on Algebraic and Geometric Topology July 29, 2018 Southwest Jiaotong University, Chengdu Wenyuan Yang Genericity of contracting elements July 29, / 44

2 Motivations: mapping class groups Elements in Mapping class groups Let Σ be a closed orientable surface of genus bigger than 2. Mapping class groups: Mod(Σ) = π 0 (Homeo + (Σ)). Teichmuller space T (Σ) = { isotopic classes of hyperbolic metrics on Σ}, endowed with the Teichmuller metric. Mod(Σ) acts isometrically and properly on T (Σ), and the quotient space T (Σ)/Mod(Σ) is the moduli space of Σ which is non-compact but of finite volume. Theorem (Nielsen-Thurston) Every mapping class [φ] Mod(Σ) is either periodic, or pseudo-anosov, or reducible. Wenyuan Yang Genericity of contracting elements July 29, / 44

3 Motivations: mapping class groups Elements in Mapping class groups Let Σ be a closed orientable surface of genus bigger than 2. Mapping class groups: Mod(Σ) = π 0 (Homeo + (Σ)). Teichmuller space T (Σ) = { isotopic classes of hyperbolic metrics on Σ}, endowed with the Teichmuller metric. Mod(Σ) acts isometrically and properly on T (Σ), and the quotient space T (Σ)/Mod(Σ) is the moduli space of Σ which is non-compact but of finite volume. Theorem (Nielsen-Thurston) Every mapping class [φ] Mod(Σ) is either periodic, or pseudo-anosov, or reducible. Wenyuan Yang Genericity of contracting elements July 29, / 44

4 Motivations: mapping class groups Pseudo-Anosov elements in 3d topology A folkfore conjecture says that most elements in Mod(Σ) are pseudo-anosov. This is supported by many evidence, e.g.: Theorem (Thurston s hyperbolization thm for mapping tori) Let [φ] Mod(Σ). Then the mapping torus Σ [0, 1]/(x, 0) (φ(x), 1) admits a hyperbolic metric iff φ is pseudo-anosov. Theorem (Agol, 2012; virtual fibering) Every closed hyperbolic manifold admits a finite cover which is the mapping torus of the above form. Wenyuan Yang Genericity of contracting elements July 29, / 44

5 Motivations: mapping class groups Pseudo-Anosov elements in 3d topology A folkfore conjecture says that most elements in Mod(Σ) are pseudo-anosov. This is supported by many evidence, e.g.: Theorem (Thurston s hyperbolization thm for mapping tori) Let [φ] Mod(Σ). Then the mapping torus Σ [0, 1]/(x, 0) (φ(x), 1) admits a hyperbolic metric iff φ is pseudo-anosov. Theorem (Agol, 2012; virtual fibering) Every closed hyperbolic manifold admits a finite cover which is the mapping torus of the above form. Wenyuan Yang Genericity of contracting elements July 29, / 44

6 Motivations: mapping class groups Quantifying genericity in random walk Writing down randomly a word of n letters, whether the probability of words being pseudo-anosov elements goes to 1 as n? Theorem (Rivin, Maher; 2011) A random word in Mod(Σ) tends to be a pseudo-anosov element with probability 1 as n. Remark This is generalized by Sisto (2011) for a much general class of groups called acylindrically hyperbolic groups, where generic elements are contracting elements. Wenyuan Yang Genericity of contracting elements July 29, / 44

7 Motivations: mapping class groups Quantifying genericity in random walk Writing down randomly a word of n letters, whether the probability of words being pseudo-anosov elements goes to 1 as n? Theorem (Rivin, Maher; 2011) A random word in Mod(Σ) tends to be a pseudo-anosov element with probability 1 as n. Remark This is generalized by Sisto (2011) for a much general class of groups called acylindrically hyperbolic groups, where generic elements are contracting elements. Wenyuan Yang Genericity of contracting elements July 29, / 44

8 Motivations: mapping class groups Quantifying genericity in word metric Conjecture (Farb, 2006) Fix a generating set S on Mod(Σ) and let d be the word metric. Consider the ball N(1, n) = {g G d(1, g) n} around the identity of radius n. Whether the following holds as n. {g N(1, n) g is pseudo-anosov} N(1, n) 1? Wenyuan Yang Genericity of contracting elements July 29, / 44

9 Motivations: mapping class groups Quantifying genericity in word metric (cont d) Some partial results: Theorem 1 [Caruso-Wiest, 2013] Pseudo-Anosov elements are generic in braid groups for Garside generating sets. 2 [Cumplido-Wiest; 2017] For word metric, a positive proportion of elements in N(1, n) of mapping class groups are pseudo-anosov. 3 [Y. 2017] We show this is also true for any subgroup with at least one pseudo-anosov element. Another related result concerns the genericity of elements in subgroups. Theorem (Gekhtman-Taylor-Tiozzo, 2016) For word metric, the generic elements in a hyperbolic subgroup of mapping class groups are pseudo-anosov. Wenyuan Yang Genericity of contracting elements July 29, / 44

10 Motivations: mapping class groups Quantifying genericity in Teichmuller metric Denote N(o, n) = {g Mod(Σ) d(o, go) n}, where o is a basepoint in T (Σ) and d is the Teichmuller metric. Theorem (Maher, 2010) {g N(o, n) g is pseudo-anosov} N(o, n) 1. However, it was unknown whether the above convergence happens exponentially fast: there exists ɛ > 0 such that {g N(o, n) g is non-pseudo-anosov} N(o, n) exp( ɛ n). Wenyuan Yang Genericity of contracting elements July 29, / 44

11 Motivations: mapping class groups Quantifying genericity in Teichmuller metric Denote N(o, n) = {g Mod(Σ) d(o, go) n}, where o is a basepoint in T (Σ) and d is the Teichmuller metric. Theorem (Maher, 2010) {g N(o, n) g is pseudo-anosov} N(o, n) 1. However, it was unknown whether the above convergence happens exponentially fast: there exists ɛ > 0 such that {g N(o, n) g is non-pseudo-anosov} N(o, n) exp( ɛ n). Wenyuan Yang Genericity of contracting elements July 29, / 44

12 Contracting elements and genericity problem Contracting elements, and examples Outline 1 Motivations: mapping class groups 2 Contracting elements and genericity problem Contracting elements, and examples Counting setup for genericity problem 3 Statistically convex-cocompact actions Convex-cocompact actions Defintions and examples Purely exponential growth 4 Genericity of contracting elements Genericity of pseudo-anosov elements Main results and corollaries 5 Ingredients in the proof of Main Theorem Wenyuan Yang Genericity of contracting elements July 29, / 44

13 Contracting elements and genericity problem Contracting elements, and examples Contracting subsets Let (Y, d) be a geodesic metric space. A subset X is called contracting if any ball missing X has a uniform bounded projection to X : there exists C > 0 such that if a metric ball B X = then Diam(Proj X (B)) C. Geodesics in trees are contracting: we can take C = 0. Wenyuan Yang Genericity of contracting elements July 29, / 44

14 Contracting elements and genericity problem Contracting elements, and examples Contracting elements Consider a proper action of a group G on (Y, d). An element g G is called contracting if for some basepoint o Y, the map n Z g n o is a quasi-isometric embedding map: λ 1, c > 0 1 λ n m c d(g n o, g m o) λ n m + c and the image g o is a contracting subset. N.B.: A quasi-isometric embedding of the integers Z is also called a quasi-geodesic. Wenyuan Yang Genericity of contracting elements July 29, / 44

15 Contracting elements and genericity problem Contracting elements, and examples Contracting elements in negatively curved spaces A geodesic metric space Y is called δ-hyperbolic if any geodesic triangle is δ thinner than the comparison triangle in a tree. In δ-hyperbolic spaces, (quasi-)geodesics are contracting. An isometry g is called loxodromic (or hyperbolic) if for some basepoint o Y, the map n Z g n o is a quasi-isometric embedding. Thus, a loxodromic element is contracting. Wenyuan Yang Genericity of contracting elements July 29, / 44

16 Contracting elements and genericity problem Contracting elements, and examples Contracting elements in negatively curved groups A finitely generated group is called hyperbolic, if it acts properly and cocompactly on a δ-hyperbolic space. A hyperbolic group consists of finite order elements and loxodromic elements. Thus, every infinite order element is contracting. Furthermore, in the class of relatively hyperbolic groups, hyperbolic elements are contracting with respect to the word metric. Wenyuan Yang Genericity of contracting elements July 29, / 44

17 Contracting elements and genericity problem Contracting elements, and examples Some facts of contracting property The contracting property can be thought of as a negative curvature property around a subset. Lemma (Thin-triangle property) Let (αβγ) be a geodesic triangle. If γ is contracting, then γ lies in a uniform neighborhood of α β. Remark! It might not hold that α is contained in some nbhd of β γ. Lemma (Morse property) Let X be a contracting subset. Then any quasi-geodesic with two endpoints in X lies in a uniform neighborhood of X. Of course, geodesics in Euclidean planes, half-planes or sectors are generally not contracting. Wenyuan Yang Genericity of contracting elements July 29, / 44

18 Contracting elements and genericity problem Contracting elements, and examples Some facts of contracting property The contracting property can be thought of as a negative curvature property around a subset. Lemma (Thin-triangle property) Let (αβγ) be a geodesic triangle. If γ is contracting, then γ lies in a uniform neighborhood of α β. Remark! It might not hold that α is contained in some nbhd of β γ. Lemma (Morse property) Let X be a contracting subset. Then any quasi-geodesic with two endpoints in X lies in a uniform neighborhood of X. Of course, geodesics in Euclidean planes, half-planes or sectors are generally not contracting. Wenyuan Yang Genericity of contracting elements July 29, / 44

19 Contracting elements and genericity problem Contracting elements, and examples An isometry g of Y is called hyperbolic if it acts by translation on a geodesic called axis. (Bestvina-Fujiwara) if the axis does not bound a Euclidean half-plane, then it is contracting. In this case, g is called a rank-1 (hyperbolic) isometry. So, rank-1 elements are contracting. Wenyuan Yang Genericity of contracting elements July 29, / 44 Contracting elements in CAT(0) spaces A geodesic metric space Y is called CAT(0) if any geodesic triangle is 0-thinner than the comparison triangle in E 2.

20 Contracting elements and genericity problem Contracting elements, and examples An isometry g of Y is called hyperbolic if it acts by translation on a geodesic called axis. (Bestvina-Fujiwara) if the axis does not bound a Euclidean half-plane, then it is contracting. In this case, g is called a rank-1 (hyperbolic) isometry. So, rank-1 elements are contracting. Wenyuan Yang Genericity of contracting elements July 29, / 44 Contracting elements in CAT(0) spaces A geodesic metric space Y is called CAT(0) if any geodesic triangle is 0-thinner than the comparison triangle in E 2.

21 Contracting elements and genericity problem Contracting elements, and examples CAT(0) groups: RAAG and RACG Let Γ be a finite simplical graph. The right-angled Artin group A(Γ) is given by the following presentation: G = V (Γ) v 1 v 2 = v 2 v 1 (v 1, v 2 ) E(Γ) (1) The right-angled Coxeter group C(Γ) is given by: G = V (Γ) v 2 = 1, v V (Γ), v 1 v 2 = v 2 v 1 (v 1, v 2 ) E(Γ) (2) Theorem [Behrstock-Charney, 12] Either a RAAG is a direct product or it contains rank-1 elements for the action on the universal cover of Salvetti complex. [Behrstock-Hagen-Sisto, 17] Either a RACG is virtually a direct product, or it contain rank-1 elements for the action on the Davis complex. Wenyuan Yang Genericity of contracting elements July 29, / 44

22 Contracting elements and genericity problem Contracting elements, and examples CAT(0) groups: RAAG and RACG Let Γ be a finite simplical graph. The right-angled Artin group A(Γ) is given by the following presentation: G = V (Γ) v 1 v 2 = v 2 v 1 (v 1, v 2 ) E(Γ) (1) The right-angled Coxeter group C(Γ) is given by: G = V (Γ) v 2 = 1, v V (Γ), v 1 v 2 = v 2 v 1 (v 1, v 2 ) E(Γ) (2) Theorem [Behrstock-Charney, 12] Either a RAAG is a direct product or it contains rank-1 elements for the action on the universal cover of Salvetti complex. [Behrstock-Hagen-Sisto, 17] Either a RACG is virtually a direct product, or it contain rank-1 elements for the action on the Davis complex. Wenyuan Yang Genericity of contracting elements July 29, / 44

23 Contracting elements and genericity problem Contracting elements, and examples Mapping class groups: pseudo-anosov elements One of the goals of this study is to understand the asymptotics of mapping class groups from a point of view of coarse geometry. Theorem (Minsky, 1997) Every pseudo-anosov element is contracting with resepct to Teichmuller metric. Wenyuan Yang Genericity of contracting elements July 29, / 44

24 Contracting elements and genericity problem Counting setup for genericity problem Outline 1 Motivations: mapping class groups 2 Contracting elements and genericity problem Contracting elements, and examples Counting setup for genericity problem 3 Statistically convex-cocompact actions Convex-cocompact actions Defintions and examples Purely exponential growth 4 Genericity of contracting elements Genericity of pseudo-anosov elements Main results and corollaries 5 Ingredients in the proof of Main Theorem Wenyuan Yang Genericity of contracting elements July 29, / 44

25 Contracting elements and genericity problem Counting setup for genericity problem Growth function and growth rate Recall that a group G acts properly on a geodesic metric space (Y, d). Fix a basepoint o Y. Denote N(o, n) = {g G d(o, go) n}. The function n N(o, n) is called growth function of the action. Look at the following asymptotic quantity called critical exponent (or growth rate): δ G = lim sup n log N(o, n). n In what follows, we always assume that δ G <. Wenyuan Yang Genericity of contracting elements July 29, / 44

26 Contracting elements and genericity problem Counting setup for genericity problem Growth rate in disguise 1 [Manning, 79] If G is the π 1 of a compact Riemannian manifold with pinched negative curature, then δ G = volume entropy = topological entropy. 2 [Coornaert, 93]If G is a hyperbolic group, then δ G = Hausdorff dimension of the Gromov boundary of G. 3 With L. Potyagailo, we generalize this to a relatively hyperbolic group, for the Floyd and Bowditch boundary of G. 4 In G = Mod(Σ), δ G = 6g 6 is the entropy of the Teichmuller geodesic flow on the Moduli space. Wenyuan Yang Genericity of contracting elements July 29, / 44

27 Contracting elements and genericity problem Counting setup for genericity problem Exponential genericity Problem (Genericity problem) [Genericity]Whether the following holds as n : {g N(o, n) g is contracting} N(o, n) 1? [Exponential genericity] If it does, does the convergence happen exponentially quick: there exists ɛ > 0 such that {g N(o, n) g is non-contracting} N(o, n) exp( ɛ n). Wenyuan Yang Genericity of contracting elements July 29, / 44

28 Contracting elements and genericity problem Counting setup for genericity problem Reformulation: purely exponential growth The action of G on X has purely exponential growth (PEG): there exists a constant C > 0 such that C 1 N(o, n) exp(δ G n) C. Assuming that G has PEG, then the set of contracting elements is exponentially generic iff the the growth rate of non-contracting elements is strictly less than the whole growth rate. Wenyuan Yang Genericity of contracting elements July 29, / 44

29 Statistically convex-cocompact actions Convex-cocompact actions Outline 1 Motivations: mapping class groups 2 Contracting elements and genericity problem Contracting elements, and examples Counting setup for genericity problem 3 Statistically convex-cocompact actions Convex-cocompact actions Defintions and examples Purely exponential growth 4 Genericity of contracting elements Genericity of pseudo-anosov elements Main results and corollaries 5 Ingredients in the proof of Main Theorem Wenyuan Yang Genericity of contracting elements July 29, / 44

30 Statistically convex-cocompact actions Convex-cocompact actions Convex-cocompact Kleinian groups A Kleinian group G is a discrete subgroup of PSL(2, C), the isometry group of H 3. The G-orbits in H 3 accumulate to a subset ΛG of the sphere S 2 called limit set of G, which is the minimal G-invariant closed subset. Consider the convex hull CH(ΛG) of the limit set. Then G is called convex-cocompact if the quotient CH(ΛG)/G is compact. 1 A convex-cocompact Kleinian group is a hyperbolic group, since CH(ΛG) is a convex subset so it is hyperbolic. 2 Compact hyperbolic 3-manifolds with totally geodesic boundary is uniformized by a convex-cocompact Kleinian group. Wenyuan Yang Genericity of contracting elements July 29, / 44

31 Statistically convex-cocompact actions Convex-cocompact actions Convex-cocompact Kleinian groups A Kleinian group G is a discrete subgroup of PSL(2, C), the isometry group of H 3. The G-orbits in H 3 accumulate to a subset ΛG of the sphere S 2 called limit set of G, which is the minimal G-invariant closed subset. Consider the convex hull CH(ΛG) of the limit set. Then G is called convex-cocompact if the quotient CH(ΛG)/G is compact. 1 A convex-cocompact Kleinian group is a hyperbolic group, since CH(ΛG) is a convex subset so it is hyperbolic. 2 Compact hyperbolic 3-manifolds with totally geodesic boundary is uniformized by a convex-cocompact Kleinian group. Wenyuan Yang Genericity of contracting elements July 29, / 44

32 Statistically convex-cocompact actions Convex-cocompact actions Convex-cocompact subgroups in Mapping class groups A subset S in a geodesic metric space (Y, d) is M-quasiconvex for some M > 0 if for any two points x, y S some geodesic [x, y] is contained in the M-neighbourhood of S. Theorem (Farb-Mosher, 2002) A finitely generated subgroup G of mapping class groups Mod(Σ) is convex-cocompact iff one of the following holds: 1 there exists some quasiconvex orbit in Teichmuller spaces; 2 every orbit is quasiconvex in Teichmuller spaces; 3 G is a hyperbolic group acting co-compactly on some weak hull of the limit set. Wenyuan Yang Genericity of contracting elements July 29, / 44

33 Statistically convex-cocompact actions Convex-cocompact actions More on convex-cocompact actions Theorem (Farb-Mosher, 2002; Hamenstadt, 2005) Let G be a convex-cocompact subgroup in Mod(Σ). 1 Every infinite order element is pseudo-anosov. 2 The natural extension Γ G of π 1 (Σ, p) with quotient G is a hyperbolic group: 1 π 1 (Σ, p) Γ G G 1. Remark (Open problems on convex-cocompact subgroups: ) Whether there exists one-ended, convex-cocompact subgroups? So far, we only know examples of free groups. Whether there exists one-ended, purely pseudo-anosov subgroups in Mod(Σ) (i.e.: every non-trivial element is pseudo-anosov). Wenyuan Yang Genericity of contracting elements July 29, / 44

34 Statistically convex-cocompact actions Convex-cocompact actions Beyond convex-cocompact actions? It is a well-known research problem to develop a generalization of convex-cocompact actions in mapping class groups. Some attempts in Mod(Σ): Mosher proposed a definition of geometrically finite subgroups; Durham-Hagen-Sisto s definition modeled on Hierarchically hyperbolic spaces. The most successful generalization is geometrically finite Kleinian groups, in the setting of Riemannian manifolds. Wenyuan Yang Genericity of contracting elements July 29, / 44

35 Statistically convex-cocompact actions Convex-cocompact actions geometrically finite Kleinain groups The π 1 (Σ) acts properly on a convex space H 2, but the action is not co-compact; it acts co-compactly on H 2 π 1 C, but which is not convex. Wenyuan Yang Genericity of contracting elements July 29, / 44

36 Statistically convex-cocompact actions Defintions and examples Outline 1 Motivations: mapping class groups 2 Contracting elements and genericity problem Contracting elements, and examples Counting setup for genericity problem 3 Statistically convex-cocompact actions Convex-cocompact actions Defintions and examples Purely exponential growth 4 Genericity of contracting elements Genericity of pseudo-anosov elements Main results and corollaries 5 Ingredients in the proof of Main Theorem Wenyuan Yang Genericity of contracting elements July 29, / 44

37 Statistically convex-cocompact actions Defintions and examples Statistically convex-cocompact actions [S.C.C.] For constants M 2 M 1 > 0, define the concave region O M1,M 2 to be the set of elements g G such that the interior of some geodesic γ between B(o, M 2 ) and B(go, M 2 ) lies outside N M1 (Go). B(o, M 2) γ go B(go, M 2) o N M1 (Go) Definition (SCC actions) A group action of G on (Y, d) is called statistically convex-cocompact action if there exist some o Y and constants M 1, M 2 > 0 such that the growth rate of O M1,M 2 is strictly less than that of Go. Wenyuan Yang Genericity of contracting elements July 29, / 44

38 Statistically convex-cocompact actions Defintions and examples Examples of SCC actions The class of SCC actions includes: Any proper and cocompact action: O M1,M 2 = ; geometrically finite Kleinian groups: O M1,M 2 consists of the union of parabolic subgroups; geometrically finite action on Hadamard manifolds with parabolic gap property introduced by Dal bo-otal-peigné (2000). The following important result allows us to study mapping class groups via SCC actions. Theorem (Eskin-Mirzakhani-Rafi; 2012) The action of mapping class group on Teichmuller space is a SCC action. Remark Recall that Minsky proved that pseudo-anosov elements are contracting with respect to the above action. Wenyuan Yang Genericity of contracting elements July 29, / 44

39 Statistically convex-cocompact actions Defintions and examples Examples of SCC actions The class of SCC actions includes: Any proper and cocompact action: O M1,M 2 = ; geometrically finite Kleinian groups: O M1,M 2 consists of the union of parabolic subgroups; geometrically finite action on Hadamard manifolds with parabolic gap property introduced by Dal bo-otal-peigné (2000). The following important result allows us to study mapping class groups via SCC actions. Theorem (Eskin-Mirzakhani-Rafi; 2012) The action of mapping class group on Teichmuller space is a SCC action. Remark Recall that Minsky proved that pseudo-anosov elements are contracting with respect to the above action. Wenyuan Yang Genericity of contracting elements July 29, / 44

40 Statistically convex-cocompact actions Defintions and examples Statistically convex-cocompact subgroups Generalizing convex-cocompact subgroups in the sense of Farb and Mosher, we propose to study the class of subgroups admiting SCC actions on Teichmuller spaces. Lemma (Y. 2017) In mapping class groups, there exists free and non-free subgroups such that they are not convex-cocompact in the sense of Farb-Mosher, but admit SCC actions on Teichmuller spaces. Wenyuan Yang Genericity of contracting elements July 29, / 44

41 Statistically convex-cocompact actions Purely exponential growth Outline 1 Motivations: mapping class groups 2 Contracting elements and genericity problem Contracting elements, and examples Counting setup for genericity problem 3 Statistically convex-cocompact actions Convex-cocompact actions Defintions and examples Purely exponential growth 4 Genericity of contracting elements Genericity of pseudo-anosov elements Main results and corollaries 5 Ingredients in the proof of Main Theorem Wenyuan Yang Genericity of contracting elements July 29, / 44

42 Statistically convex-cocompact actions Purely exponential growth Purely exponential growth Recall that the growth rate: Theorem A (Y. 2016) δ G = lim sup n log N(o, n). n Suppose G admits a statistically convex-cocompact action on a geodesic metric space (X, d) with a contracting element. Then G has purely exponential growth: there exists a constant C > 0 such that Remark In general, the limit N(o,n) exp(δ G n) C 1 N(o, n) exp(δ G n) C. may not exist. Wenyuan Yang Genericity of contracting elements July 29, / 44

43 Statistically convex-cocompact actions Purely exponential growth Purely exponential growth Recall that the growth rate: Theorem A (Y. 2016) δ G = lim sup n log N(o, n). n Suppose G admits a statistically convex-cocompact action on a geodesic metric space (X, d) with a contracting element. Then G has purely exponential growth: there exists a constant C > 0 such that Remark In general, the limit N(o,n) exp(δ G n) C 1 N(o, n) exp(δ G n) C. may not exist. Wenyuan Yang Genericity of contracting elements July 29, / 44

44 Statistically convex-cocompact actions Purely exponential growth Corollaries 1 Right-angled Artin groups have purely exponential growth, if there exists a rank-1 element; 2 Right-angled Coxeter groups have purely exponential growth, if there exists a rank-1 element; 3 [Athreya-Bufetov-Eskin-Mirzakhani, 2011] Mod(Σ) has purely exponential growth. In fact, they proved that the limit exists: N(o, n) exp(δ G n). Wenyuan Yang Genericity of contracting elements July 29, / 44

45 Genericity of contracting elements Genericity of pseudo-anosov elements Outline 1 Motivations: mapping class groups 2 Contracting elements and genericity problem Contracting elements, and examples Counting setup for genericity problem 3 Statistically convex-cocompact actions Convex-cocompact actions Defintions and examples Purely exponential growth 4 Genericity of contracting elements Genericity of pseudo-anosov elements Main results and corollaries 5 Ingredients in the proof of Main Theorem Wenyuan Yang Genericity of contracting elements July 29, / 44

46 Genericity of contracting elements Main results and corollaries Outline 1 Motivations: mapping class groups 2 Contracting elements and genericity problem Contracting elements, and examples Counting setup for genericity problem 3 Statistically convex-cocompact actions Convex-cocompact actions Defintions and examples Purely exponential growth 4 Genericity of contracting elements Genericity of pseudo-anosov elements Main results and corollaries 5 Ingredients in the proof of Main Theorem Wenyuan Yang Genericity of contracting elements July 29, / 44

47 Genericity of contracting elements Main results and corollaries Main results for SCC actions Theorem A (Y. 2017: SCC action) Suppose G admits a SCC action on a geodesic metric space (X, d) with a contracting element. Then contracting elements are exponentially generic: there exists ɛ > 0 such that {g N(o, n) g is non-contracting} N(o, n) exp( ɛ n). Wenyuan Yang Genericity of contracting elements July 29, / 44

48 Genericity of contracting elements Main results and corollaries Corollary 1: mapping class groups Since mapping class groups act on Teichmuller spaces by SCC action (Eskin-Mirzakhani-Rafi; 2012) and contain contracting elements (Minsky, 1997), we have: Corollary (Mapping class groups) Pseudo-Anosov elements in Mapping class groups are exponentially generic in Teichmuller metric. Remark The genericity part was proved by Maher; here we strength the convergence speed to get exponentially genericity. Independently, this was also claimed by Masur and Dowdall around the same time. Wenyuan Yang Genericity of contracting elements July 29, / 44

49 Genericity of contracting elements Main results and corollaries Corollary 2 Recall that RAAGs/RACGs act cocompactly on some canoncial nonpositively curved cube complexes: Corollary (CAT(0) groups) With respect to the CAT(0) metric, the set of rank-1 elements, if exists, is exponentially generic in right-angled Artin groups. With respect to the CAT(0) metric, the set of rank-1 elements, if exists, is exponentially generic in right-angled Artin groups. Corollary (Relatively hyperbolic groups) The set of hyperbolic elements in a relatively hyperbolic group is exponentially generic with respect to the word metric. Wenyuan Yang Genericity of contracting elements July 29, / 44

50 Genericity of contracting elements Main results and corollaries Corollary 2 Recall that RAAGs/RACGs act cocompactly on some canoncial nonpositively curved cube complexes: Corollary (CAT(0) groups) With respect to the CAT(0) metric, the set of rank-1 elements, if exists, is exponentially generic in right-angled Artin groups. With respect to the CAT(0) metric, the set of rank-1 elements, if exists, is exponentially generic in right-angled Artin groups. Corollary (Relatively hyperbolic groups) The set of hyperbolic elements in a relatively hyperbolic group is exponentially generic with respect to the word metric. Wenyuan Yang Genericity of contracting elements July 29, / 44

51 Genericity of contracting elements Main results and corollaries Main results for proper actions For general proper actions, we have the following. Theorem B (Y. 2017: proper action) Suppose G admits a proper action with purely exponential growth on a geodesic metric space (X, d) with a contracting element. Then the set of contracting elements is generic. Remark We do not know whether the genericity can be improved to be the exponential genericity. Wenyuan Yang Genericity of contracting elements July 29, / 44

52 Genericity of contracting elements Main results and corollaries Corollary 1 By McCarthy and Papadopoulos, a sufficiently large subgroup contains at least two independent pseudo-anosov elements. Corollary (Mapping class groups) If a sufficiently large subgroup has purely exponential growth, then the set of pseudo-anosov elements is generic. Question Under which condition, sufficiently large subgroups have purely exponential growth? Wenyuan Yang Genericity of contracting elements July 29, / 44

53 Genericity of contracting elements Main results and corollaries Corollary 1 By McCarthy and Papadopoulos, a sufficiently large subgroup contains at least two independent pseudo-anosov elements. Corollary (Mapping class groups) If a sufficiently large subgroup has purely exponential growth, then the set of pseudo-anosov elements is generic. Question Under which condition, sufficiently large subgroups have purely exponential growth? Wenyuan Yang Genericity of contracting elements July 29, / 44

54 Genericity of contracting elements Main results and corollaries Corollary 2 The above question is completely answered in Riemanian manifolds. Theorem (Roblin, 2003) Purely exponential growth of a proper action on CAT(-1) spaces is characterized by the finiteness of Bowen-Margulis-Sullivan measures. As a corollary, we obtain: Corollary (CAT(-1) groups) The set of hyperbolic elements in a proper group action on CAT(-1) space with finite Bowen-Margulis-Sullivan measure is generic. Wenyuan Yang Genericity of contracting elements July 29, / 44

55 Genericity of contracting elements Main results and corollaries Corollary 2 The above question is completely answered in Riemanian manifolds. Theorem (Roblin, 2003) Purely exponential growth of a proper action on CAT(-1) spaces is characterized by the finiteness of Bowen-Margulis-Sullivan measures. As a corollary, we obtain: Corollary (CAT(-1) groups) The set of hyperbolic elements in a proper group action on CAT(-1) space with finite Bowen-Margulis-Sullivan measure is generic. Wenyuan Yang Genericity of contracting elements July 29, / 44

56 Ingredients in the proof of Main Theorem Proof of the main results 1 First establish a growth tightness result. This gives a criterion on which subsets are exponentially small: exp(nδ) for some δ < δ G ; 2 Prove then that conjugacy classes of non-contracting elements are exponentially small. 3 Using projection complex due to Bestvina-Bromberg-Fujiwara, we show that the non-contracting elements can be decomposed as almost geodesic form: g = k h k 1 so that d(o, go) 2 d(o, ko) + d(o, ho), where h is a shortest conjugacy representative of g. 4 The goal is to count the non-contracting elements g such that d(o, go) = n the number of h is at most exp(δd(o, ho)) for δ < δ G, the number of k is at most exp( δ G 2 (n d(o, ho))). Hence, the growth rate of non-contracting g is max{δ, δ G /2} < δ G Wenyuan Yang Genericity of contracting elements July 29, / 44

57 Ingredients in the proof of Main Theorem Proof of the main results 1 First establish a growth tightness result. This gives a criterion on which subsets are exponentially small: exp(nδ) for some δ < δ G ; 2 Prove then that conjugacy classes of non-contracting elements are exponentially small. 3 Using projection complex due to Bestvina-Bromberg-Fujiwara, we show that the non-contracting elements can be decomposed as almost geodesic form: g = k h k 1 so that d(o, go) 2 d(o, ko) + d(o, ho), where h is a shortest conjugacy representative of g. 4 The goal is to count the non-contracting elements g such that d(o, go) = n the number of h is at most exp(δd(o, ho)) for δ < δ G, the number of k is at most exp( δ G 2 (n d(o, ho))). Hence, the growth rate of non-contracting g is max{δ, δ G /2} < δ G Wenyuan Yang Genericity of contracting elements July 29, / 44

58 Ingredients in the proof of Main Theorem Proof of the main results 1 First establish a growth tightness result. This gives a criterion on which subsets are exponentially small: exp(nδ) for some δ < δ G ; 2 Prove then that conjugacy classes of non-contracting elements are exponentially small. 3 Using projection complex due to Bestvina-Bromberg-Fujiwara, we show that the non-contracting elements can be decomposed as almost geodesic form: g = k h k 1 so that d(o, go) 2 d(o, ko) + d(o, ho), where h is a shortest conjugacy representative of g. 4 The goal is to count the non-contracting elements g such that d(o, go) = n the number of h is at most exp(δd(o, ho)) for δ < δ G, the number of k is at most exp( δ G 2 (n d(o, ho))). Hence, the growth rate of non-contracting g is max{δ, δ G /2} < δ G Wenyuan Yang Genericity of contracting elements July 29, / 44

59 Ingredients in the proof of Main Theorem Proof of the main results 1 First establish a growth tightness result. This gives a criterion on which subsets are exponentially small: exp(nδ) for some δ < δ G ; 2 Prove then that conjugacy classes of non-contracting elements are exponentially small. 3 Using projection complex due to Bestvina-Bromberg-Fujiwara, we show that the non-contracting elements can be decomposed as almost geodesic form: g = k h k 1 so that d(o, go) 2 d(o, ko) + d(o, ho), where h is a shortest conjugacy representative of g. 4 The goal is to count the non-contracting elements g such that d(o, go) = n the number of h is at most exp(δd(o, ho)) for δ < δ G, the number of k is at most exp( δ G 2 (n d(o, ho))). Hence, the growth rate of non-contracting g is max{δ, δ G /2} < δ G Wenyuan Yang Genericity of contracting elements July 29, / 44

60 Ingredients in the proof of Main Theorem Conjecture Mapping class groups contain a contracting element for the action on the Cayley graph. If this is true, this would imply that Farb s conjecture is true: pseudo-anosov elements are exponentially generic for word metric. Thank you for your attention! Wenyuan Yang Genericity of contracting elements July 29, / 44