Short hierarchically hyperbolic groups II: quotients and the Hopf property for Artin groups

TL;DR

This paper proves that most large, hyperbolic Artin groups are Hopfian and residually finite, and shows that many quotients of the five-holed sphere mapping class group are hierarchically hyperbolic, using a Dehn-filling-like method.

Contribution

It introduces a Dehn-filling-like procedure for short hierarchically hyperbolic groups and applies it to establish Hopfian and residual finiteness properties of Artin groups.

Findings

Most Artin groups of large, hyperbolic type are Hopfian.

Under certain conditions, all such Artin groups are residually finite.

Most quotients of the five-holed sphere mapping class group are hierarchically hyperbolic.

Abstract

We prove that most Artin groups of large and hyperbolic type are Hopfian, meaning that every self-epimorphism is an isomorphism. The class covered by our result is generic, in the sense of Goldsborough-Vaskou. Moreover, assuming the residual finiteness of certain hyperbolic groups with an explicit presentation, we get that all large and hyperbolic type Artin groups are residually finite. We also show that most quotients of the five-holed sphere mapping class group are hierarchically hyperbolic, up to taking powers of the normal generators of the kernels. The main tool we use to prove both results is a Dehn-filling-like procedure for short hierarchically hyperbolic groups (these also include e.g. non-geometric 3-manifolds, and triangle- and square-free RAAGs).