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Article Dans Une Revue Proceedings of the Royal Society of Edinburgh: Section A, Mathematics Année : 2021

Dehn Filling Dehn Twists

Résumé

Abstract 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 3 g − 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 .

Dates et versions

hal-03951294 , version 1 (23-01-2023)

Identifiants

Citer

François Dahmani, Mark Hagen, Alessandro Sisto. Dehn Filling Dehn Twists. Proceedings of the Royal Society of Edinburgh: Section A, Mathematics, 2021, 151 (1), pp.28-51. ⟨10.1017/prm.2020.1⟩. ⟨hal-03951294⟩

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