Finite groups of untwisted outer automorphisms of RAAGs

TL;DR

This paper proves that all finite subgroups of a certain automorphism subgroup of right-angled Artin groups fix points in a contractible space, leading to finiteness results on conjugacy classes of these subgroups.

Contribution

It generalizes fixed point results from free groups to right-angled Artin groups, showing finite subgroups of $U^0(A_{\Gamma})$ fix points in the associated space.

Findings

Finite subgroups of $U^0(A_{\Gamma})$ fix points in $K_{\Gamma}$

There are finitely many conjugacy classes of finite subgroups in $U^0(A_{\Gamma})$

Extension of fixed point properties from free groups to RAAGs

Abstract

For any right-angled Artin group $A_{\Gamma}$, Charney--Stambaugh--Vogtmann showed that the subgroup $U^0(A_{\Gamma}) \leq\text{Out}(A_{\Gamma})$ generated by Whitehead automorphisms and inversions acts properly and cocompactly on a contractible space $K_{\Gamma}$. In the present paper we show that any finite subgroup of $U^0(A_{\Gamma})$ fixes a point of $K_{\Gamma}$. This generalizes the fact that any finite subgroup of $\text{Out}(F_n)$ fixes a point of Outer Space, and implies that there are only finitely many conjugacy classes of finite subgroups in $U^0(A_{\Gamma})$.