Dehn filling Dehn twists

Abstract

AbstractLet$\Sigma _{g,p}$be the genus–goriented surface withppunctures, with eitherg> 0 orp> 3. We show that$MCG(\Sigma _{g,p})/DT$is acylindrically hyperbolic whereDTis 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 suitableK.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 groupGon a hyperbolic graphX. We give conditions ensuring that the graphX/Nis again hyperbolic and various properties of the action ofGonXpersist for the action ofG/NonX/N.