Convex cocompactness and stability in mapping class groups

By Samuel J. Taylor. 2014

82 citations recorded by the source corpus.

We introduce a strong notion of quasiconvexity in finitely generated groups, which we call stability. Stability agrees with quasiconvexity in hyperbolic groups and is preserved under quasi-isometry for finitely generated groups. We show that the stable subgroups of mapping class groups are precisely the convex cocompact subgroups. This generalizes a well-known result of Behrstock and is related to questions asked by Farb and Mosher and by Farb. 20F65, 51H05; 57M07, 30F60

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5a7497f6c4d8e4c3eeb63c33ce9e1c4055675183

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

On strongly quasiconvex subgroups (2017, background): “ the Cayley graph Γ(G,S). We say H is stable in G if Γ(H,T) is stable in Γ(G,S). We remark that stable subgroups were proved to be independent of the choice of ﬁnite generating sets (see Section 3 in [DT15b]). Deﬁnition 4.5. Let G be a ﬁnite generated group and H a subgroup of G. We say H is quasico”
Hierarchically hyperbolic spaces II: Combination theorems and the distance formula (2015, background): “In [DT14], the authors characterize stable subgroups of the mapping class group; it is reasonable to ask for a generalization of their results to hierarchically hyperbolic groups.”
Largest acylindrical actions and Stability in hierarchically hyperbolic groups (2017, methodology): “is a universal acylindrical action. 2.3. Stability. Stability isstrong coarse convexity property which generalizes quasiconvexity in hyperbolic spaces and convex cocompactness in mapping class groups [DT15]. In the general context of metric spaces, it is essentially the familiar Morse property gener”
Boundaries and automorphisms of hierarchically hyperbolic spaces (2016, background): “ivalent characterizations, which we state in the following theorem-deﬁnition with parts due variously to Farb-Mosher [FM02], Hamenstädt [Ham05], Kent-Leininger [KL08], and the ﬁrst author with Taylor [DT15]: Theorem 5.12. A subgroup H ă MCGpSq is convex cocompact if it satisﬁes any of the following ”
Morse Boundaries of Proper Geodesic Metric Spaces (2015, unknown): “ce from f(γ). (Equivalently, given N∈Mthere exists an N′ ∈Msuch that for every N-Morse geodesic ray γ, f(γ) is an N′-Morse quasi-geodesic.) For example, if X⊂Y is stable as deﬁned by Durham–Taylor in [8], then the associated quasi-isometric embedding X →Y is always Morse preserving. If f: X→Y is a q”
Hyperbolicities in CAT(0) cube complexes (2017, background): “e following statements are equivalent: • G is acylindrically hyperbolic; • G contains an inﬁnite stable subgroup of inﬁnite index; • G contains an inﬁnite Morse subgroup of inﬁnite index. Recall from [DT15] that a subgroup H in a ﬁnitely generated group G is stable if, for any constants A ≥ 1 and B ”