Surface subgroups of mapping class groups

Reid, Alan W.

doi:10.1090/pspum/074/2264545

Cited by 7 publications

(4 citation statements)

References 27 publications

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“…In [30], Reid posed the question of whether convex-cocompact subgroups of MCG(Σ g,p ) are separable. Recall that a subgroup H < G is separable if for every x ∈ G − H there exists a finite group F a surjective homomorphism φ : G → F with φ(x) / ∈ φ(H).…”

Section: Residual Properties Of Mapping Class Groups In Low Complexitymentioning

confidence: 99%

Dehn filling Dehn twists

Dahmani¹,

Hagen²,

Sisto³

2020

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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Let $\Sigma _{g,p}$ be the genus–g oriented surface with p punctures, with either g > 0 or p > 3. We show that $MCG(\Sigma _{g,p})/DT$ is acylindrically hyperbolic where DT is the normal subgroup of the mapping class group $MCG(\Sigma _{g,p})$ generated by $K^{th}$ powers of Dehn twists about curves in $\Sigma _{g,p}$ for suitable K. Moreover, we show that in low complexity $MCG(\Sigma _{g,p})/DT$ is in fact hyperbolic. In particular, for 3g − 3 + p ⩽ 2, we show that the mapping class group $MCG(\Sigma _{g,p})$ is fully residually non-elementary hyperbolic and admits an affine isometric action with unbounded orbits on some $L^q$ space. Moreover, if every hyperbolic group is residually finite, then every convex-cocompact subgroup of $MCG(\Sigma _{g,p})$ is separable. The aforementioned results follow from general theorems about composite rotating families, in the sense of [13], that come from a collection of subgroups of vertex stabilizers for the action of a group G on a hyperbolic graph X. We give conditions ensuring that the graph X/N is again hyperbolic and various properties of the action of G on X persist for the action of G/N on X/N.

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Section: Residual Properties Of Mapping Class Groups In Low Complexitymentioning

confidence: 99%

Dehn filling Dehn twists

Dahmani¹,

Hagen²,

Sisto³

2020

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…In [Rei06], Reid posed the question of whether convex-cocompact subgroups of M CG(Σ g,p ) are separable. Recall that a subgroup H < G is separable if for every x ∈ G − H there exists a finite group F a surjective homomorphism φ : G → F with φ(x) / ∈ φ(H).…”

Section: Residual Properties Of Mapping Class Groups In Low Complexitymentioning

confidence: 99%

Dehn filling Dehn twists

Dahmani,

Hagen,

Sisto

2018

Preprint

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Let Σg,p be the genus-g oriented surface with p punctures, with either g > 0 or p > 3. We show that M CG(Σg,p)/DT is acylindrically hyperbolic where DT is the normal subgroup of the mapping class group M CG(Σg,p) generated by K th powers of Dehn twists about curves in Σg,p for suitable K.Moreover, we show that in low complexity M CG(Σg,p)/DT is in fact hyperbolic. In particular, for 3g − 3 + p ≤ 2, we show that the mapping class group M CG(Σg,p) is fully residually non-elementary hyperbolic and admits an affine isometric action with unbounded orbits on some L q space. Moreover, if every hyperbolic group is residually finite, then every convex-cocompact subgroup of M CG(Σg,p) is separable.The aforementioned results follow from general theorems about composite rotating families, in the sense of [Dah18], that come from a collection of subgroups of vertex stabilisers for the action of a group G on a hyperbolic graph X. We give conditions ensuring that the graph X/N is again hyperbolic and various properties of the action of G on X persist for the action of G/N on X/N . ContentsHagen was supported by EPSRC grant EPSRC EP/R042187/1.

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“…In the context of mapping class groups, the closest analogues of quasiconvex subgroups of hyperbolic groups are convex‐cocompact subgroups, as defined in [14]; see Subsection 2.1 for the characterisation that we will use. In analogy with the case of hyperbolic manifolds, Reid asked whether convex‐cocompact subgroups of mapping class groups are separable [30, Question 3.5]. As far as we are aware, the only examples of convex‐cocompact subgroups known to be separable are virtually cyclic, as covered by [20].…”

Section: Introductionmentioning

confidence: 99%