Convex cocompact subgroups of mapping class groups

By Benson Farb, Lee Mosher. 2001

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We develop a theory of convex cocompact subgroups of the mapping class group MCG of a closed, oriented surface S of genus at least 2, in terms of the action on Teichmuller space. Given a subgroup G of MCG defining an extension 1 → π 1 (S) → T G → G → 1, we prove that if Γ G is a word hyperbolic group then G is a convex cocompact subgroup of MCG. When G is free and convex cocompact, called a Schottky subgroup of MCG, the converse is true as well; a semidirect product of π 1 (S) by a free group G is therefore word hyperbolic if and only if G is a Schottky subgroup of MCG. The special case when G = Z follows from Thurston's hyperbolization theorem. Schottky subgroups exist in abundance: sufficiently high powers of any independent set of pseudo-Anosov mapping classes freely generate a Schottky subgroup.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/549e3fae63325939958d0776ce5925ff63f09fe2

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Citations in the corpus

Boundaries and automorphisms of hierarchically hyperbolic spaces (2016, background): “teness. In particular, we prove: Theorem 3. Suppose that Hă MCGpSq is one of the following: (1) The standard embedding of MCGpYq for some proper subsurface Y Ă S; (2) Convex cocompact in the sense of [FM02]; (3) A ﬁnitely generated Veech group; (4) A Leininger-Reid combination subgroup [LR06]. Then ”
The geometry of right angled Artin subgroups of mapping class groups (2010, background): “In [3], Behrstock, Kleiner, Minsky and Mosher defined a partial order on Ω(K,µ, µ′) (for K sufficiently large) which is closely related to the time-order constructed in [33] (see also [4]).”
Morse Boundaries of Proper Geodesic Metric Spaces (2015, unknown): “emma 4.13. Let −−−→ [x,ξ), −−−→ [y,η) be two Morse rays in T(S). The rays −−−→ [x,ξ), −−−→ [y,η) have ﬁnite Hausdorﬀ distance in T(S) if and only if ξ= η. Proof. The forward direction is Lemma 2.4 in [9]. For the other direction we apply the result of Masur mentioned above [14]. Proposition 4.14. Th”
Statistically convex-cocompact actions of groups with contracting elements (2016, methodology): “The notion of convex-cocompact subgroups in Mod was introduced by Farb and Mosher [35].”
Hyperbolic extensions of free groups (2014, methodology): “(See the papers of Farb–Mosher [ FM1] and Kent– Farb and Mosher show that if E G is hyperbolic then G is convex cocompact.”
On strongly quasiconvex subgroups (2017, background): “isometry, and agrees with quasiconvexity when the ambient group is hyperbolic. Moreover, stable subgroups of mapping class groups are precisely the convex cocompact subgroups deﬁned by Farb-Mosher in [FM02] and the such subgroups of mapping class groups is a primary motivation for the concept of sta”
Convex cocompactness and stability in mapping class groups (2014, methodology): “(See, for example, Farb and Mosher [10], Kent and Leininger [13], Kent, Leininger and Schleimer [14], Dowdall, Kent and Leininger [5], Mangahas and Taylor [15], and Min [19].”
A Combination Theorem for Metric Bundles (2009, background): “Ki)},i= 1,··· ,n. Conversely, if Gis (strongly) hyperbolic relative to the collection {NG(Ki)},i= 1,··· ,n, then Qis convex-cocompact. Proof. Suppose that Qis convex cocompact. Then Qis hyperbolic by [FM02], [KL08]. Also, by Theorem 5.2, Q= Qi for all i. This is because Kis strongly hyperbolic relat”
A hyperbolic Out(F_n)-complex (2008, background): “ose some pseudo-Anosov homeomorphism g whose stable and unstable laminations are very close, but not equal to those of f1. This is possible by the work of Farb and Mosher on Schottky subgroups of MCG [FM02]. Furthermore, one can arrange that nontrivial powers of g are not conjugate to powers of fi’s”
On the growth of $L^2$-invariants for sequences of lattices in Lie groups (2012, background): “si-convex, i.e. after ﬁxing X∈T(Σ), there is some C>0 such that any Teichmuller geodesic segment that joins two points from the orbit F(X) lies in a C-neighborhood of F(X). Remark. Farb and Mosher [50] have shown that if φ1,...,φn are pseudo-Anosov elements of Mod(Σ) with pairwise distinct attrac”
Shadows Of Mapping Class Groups: Capturing Convex Cocompactness (2005, background): “The allure of convex cocompact mapping class groups is of a ma nifestly different nature: Farb and Mosher have shown that when S is closed, convex cocompactness for a subgroup G < Mod(S) is implied by theδ–hyperbolicity of the associated π1(S)–extension ofG—see [20].”