Fixed subgroups in Artin groups

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.