Homomorphisms between XL-type Artin groups

TL;DR

This paper investigates homomorphisms between XL-type Artin groups, establishing conditions under which these groups are hopfian, co-hopfian, and describing their automorphism groups based on graph properties.

Contribution

It provides a comprehensive analysis of homomorphisms and automorphisms of XL-type Artin groups, including characterizations for groups over complete and general graphs.

Findings

XL-type Artin groups over complete graphs are both hopfian and co-hopfian.

Characterization of co-hopfian properties for groups over graphs with labels ≥6.

Finite generation of automorphism groups when the graph has no cut-vertex.

Abstract

We study homomorphisms between XL-type Artin groups and show that, in a suitable sense, a generic Artin group is both hopfian and co-hopfian. For XL-type Artin groups over complete graphs, we describe all possible homomorphisms with sufficiently large image, and prove in particular that such groups are both hopfian and co-hopfian. For Artin groups over general graphs with all labels at least $6$, we characterise in terms of the presentation graph exactly when these groups are co-hopfian, as well as when they have a finite outer automorphism group. When in addition the presentation graph has no cut-vertex, we show that their automorphism group is finitely generated and we provide a generating set.