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arXiv · 2410.19091

Homomorphisms between XL-type Artin groups

Martín Blufstein·Alexandre Martin·Nicolas Vaskou
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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.

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Martín Blufstein, Alexandre Martin, Nicolas Vaskou. 2024-10-24. Homomorphisms between XL-type Artin groups. https://arxiv.org/abs/2410.19091

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