# Dehn filling Dehn twists

## TL;DR

This paper investigates the hyperbolic properties of certain quotients of the mapping class group of punctured surfaces, revealing conditions under which these quotients are hyperbolic or acylindrically hyperbolic, with implications for subgroup separability.

## Contribution

The paper introduces new results on the hyperbolic nature of quotients of the mapping class group by powers of Dehn twists, extending understanding of their geometric and algebraic properties.

## Key findings

- $MCG(\Sigma_{g,p})/DT$ is acylindrically hyperbolic for suitable $K$.
- In low complexity, $MCG(\Sigma_{g,p})/DT$ is hyperbolic.
- Mapping class groups are fully residually hyperbolic in certain cases.

## 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 $3g-3+p\leq 2$, 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 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 $X/N$.