Homomorphisms between XL-type Artin groups

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.