Fixed subgroups in Artin groups

Abstract

We study fixed subgroups of automorphisms of any large-type Artin group $A_{\hat{I}}“$. We define a natural subgroup ${\mathrm{Aut}}_{\hat{I}}“(A_{\hat{I}}“)$ of $\mathrm{Aut}(A_{\hat{I}}“)$, and for every ${\hat{I}}^3\in {\mathrm{Aut}}_{\hat{I}}“(A_{\hat{I}}“)$ we find the isomorphism type of $\mathrm{Fix}({\hat{I}}^3)$ and a generating set for a finite index subgroup. We show that $\mathrm{Fix}({\hat{I}}^3)$ is a finitely generated Artin group, with a uniform bound on the rank in terms of the number of vertices of $\hat{I}“$. Finally, we provide a natural geometric characterisation of the subgroup ${\mathrm{Aut}}_{\hat{I}}“(A_{\hat{I}}“)$, which informally is the maximal subgroup of $\mathrm{Aut}(A_{\hat{I}}“)$ leaving the Deligne complex of $A_{\hat{I}}“$ invariant.