Cusped spaces for hierarchically hyperbolic groups, and applications to Dehn filling quotients

TL;DR

The paper introduces a new construction for cusped spaces in relatively hyperbolic groups, applying it to study Dehn-filling quotients of groups like mapping class groups, revealing unique hyperbolic quotients with trivial outer automorphisms.

Contribution

It presents a novel construction of cusped spaces for relatively hyperbolic groups and applies it to analyze Dehn-filling quotients of mapping class groups and braid groups.

Findings

Mapping class group of a five-holed sphere has infinite hyperbolic quotients not isomorphic to others.

Quotients obtained by large powers of Dehn twists have trivial outer automorphism groups.

The construction links cusped spaces with Teichmüller metrics for various groups.

Abstract

We introduce a construction that simultaneously yields cusped spaces of relatively hyperbolic groups, and spaces quasi-isometric to Teichmueller metrics. We use this to study Dehn-filling-like quotients of various groups, among which mapping class groups of punctured spheres. In particular, we show that the mapping class group of a five-holed sphere (resp. the braid group on four strands) has infinite hyperbolic quotients (strongly) not isomorphic to hyperbolic quotients of any other given sphere mapping class group (resp. any other braid group). These quotients are obtained by modding out suitable large powers of Dehn twists, and we further argue that the corresponding quotients of the extended mapping class group have trivial outer automorphism groups. We obtain these results by studying torsion elements in the relevant quotients.