Subgroups of mapping class groups from the geometrical viewpoint

Kent, Richard P.; Leininger, Christopher J.

doi:10.1090/conm/432/08306

Cited by 32 publications

(21 citation statements)

References 59 publications

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“…A finitely generated subgroup Γ ≤ Mod(S) is convex cocompact if for some (any) x ∈ Teich(S), the Teichmüller space of the surface S, the orbit Γ · x ⊂ Teich(S) is quasiconvex with respect to the Teichmüller metric. (See the papers of Farb-Mosher [FM1] and KL4] for definitions and details). Similar to the situation described above, a subgroup Γ ≤ Mod(S) gives rise to a surface group extension 1 −→ π 1 (S) −→ E Γ −→ Γ −→ 1.…”

Section: Motivation From Surface Group Extensions and Some Previous Rmentioning

confidence: 99%

Hyperbolic extensions of free groups

Dowdall

Taylor

2017

Geom. Topol.

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Given a finitely generated subgroup Γ ≤ Out(F) of the outer automorphism group of the rank r free group F = F r , there is a corresponding free group extension 1 → F → E Γ → Γ → 1. We give sufficient conditions for when the extension E Γ is hyperbolic. In particular, we show that if all infinite order elements of Γ are atoroidal and the action of Γ on the free factor complex of F has a quasi-isometric orbit map, then E Γ is hyperbolic. As an application, we produce examples of hyperbolic F-extensions E Γ for which Γ has torsion and is not virtually cyclic. The proof of our main theorem involves a detailed study of quasigeodesics in Outer space that make progress in the free factor complex. This may be of independent interest.

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Section: Motivation From Surface Group Extensions and Some Previous Rmentioning

confidence: 99%

Hyperbolic extensions of free groups

Dowdall

Taylor

2017

Geom. Topol.

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“…Although Teichmüller space itself is in no ordinary sense negatively curved (for example, [18,21]), a driving principle in the study of the coarse geometry of T (S) is that the thick part T has many hyperboliclike features. See, for example, [14]. One manifestation of this principle is Minsky's theorem that Teichmüller geodesics which remain in the thick part of Teichmüller space are strongly contracting.…”

Section: Necessity Of Quasiconvexity In A(γ) and The Proof Of Theoremmentioning

confidence: 99%

Convex cocompactness in mapping class groups via quasiconvexity in right-angled Artin groups

Mangahas

Taylor

2016

Proc. London Math. Soc.

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Abstract. We characterize convex cocompact subgroups of mapping class groups that arise as subgroups of specially embedded right-angled Artin groups. That is, if the right-angled Artin group G < Mod(S) satisfies certain conditions that imply G is quasi-isometrically embedded in Mod(S), then a purely pseudo-Anosov H < G is convex cocompact in Mod(S) if and only if it is combinatorially quasiconvex in G. We use this criterion to construct convex cocompact subgroups of Mod(S) whose orbit maps into the curve complex have small Lipschitz constants.

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“…In [KL1,KL2], we extended Farb and Mosher's analogy, providing several characterizations of convex cocompactness borrowed from the Kleinian setting (see also Hamenstädt [H]). The analogy is an imperfect one, see [KL3] and the references there, and we point out some new imperfections here. i and yet f 1 , f 2 ∼ = F 2 is not a Schottky group.…”

Section: Introductionmentioning

confidence: 96%