### Abstract

Let be the genus-g oriented surface with p punctures, with either g > 0 or p > 3. We show that is acylindrically hyperbolic where DT is the normal subgroup of the mapping class group generated by powers of Dehn twists about curves in for suitable K.Moreover, we show that in low complexity is in fact hyperbolic. In particular, for 3g - 3 + p 1/2 2, we show that the mapping class group is fully residually non-elementary hyperbolic and admits an affine isometric action with unbounded orbits on some space. Moreover, if every hyperbolic group is residually finite, then every convex-cocompact subgroup of 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.

Original language | English |
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Journal | Proceedings of the Royal Society of Edinburgh Section A: Mathematics |

Early online date | 30 Jan 2020 |

DOIs | |

Publication status | E-pub ahead of print - 30 Jan 2020 |

### Keywords

- Dehn twists
- Mapping class groups

### ASJC Scopus subject areas

- Mathematics(all)

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## Profiles

## Alessandro Sisto

- School of Mathematical & Computer Sciences - Assistant Professor
- School of Mathematical & Computer Sciences, Mathematics - Assistant Professor

Person: Academic (Research & Teaching)

## Cite this

*Proceedings of the Royal Society of Edinburgh Section A: Mathematics*. https://doi.org/10.1017/prm.2020.1