Fixed subgroups in Artin groups

TL;DR

This paper investigates fixed subgroups of automorphisms in large-type Artin groups, characterizing their structure, generators, and geometric properties, revealing they are finitely generated Artin groups with bounded rank.

Contribution

It introduces a natural automorphism subgroup, determines fixed subgroup isomorphism types, and provides a geometric characterization of automorphisms preserving the Deligne complex.

Findings

Fixed subgroups are finitely generated Artin groups.

Uniform bound on the rank of fixed subgroups.

Automorphisms preserving the Deligne complex form a maximal subgroup.

Abstract

We study fixed subgroups of automorphisms of any large-type Artin group $A_{\Gamma}$. We define a natural subgroup $\mathrm{Aut}_\Gamma(A_\Gamma)$ of $\mathrm{Aut}(A_{\Gamma})$, and for every $\gamma \in \mathrm{Aut}_\Gamma(A_\Gamma)$ we find the isomorphism type of $\mathrm{Fix}(\gamma)$ and a generating set for a finite index subgroup. We show that $\mathrm{Fix}(\gamma)$ is a finitely generated Artin group, with a uniform bound on the rank in terms of the number of vertices of $\Gamma$. Finally, we provide a natural geometric characterisation of the subgroup $\mathrm{Aut}_\Gamma(A_\Gamma)$, which informally is the maximal subgroup of $\mathrm{Aut}(A_\Gamma)$ leaving the Deligne complex of $A_{\Gamma}$ invariant.