Finite groups of untwisted outer automorphisms of RAAGs

Abstract

For any right-angled Artin group $A_{\hat{I}}“$, Charney--Stambaugh--Vogtmann showed that the subgroup $U^0(A_{\hat{I}}“)\le \text{Out}(A_{\hat{I}}“)$ generated by Whitehead automorphisms and inversions acts properly and cocompactly on a contractible space $K_{\hat{I}}“$. In the present paper we show that any finite subgroup of $U^0(A_{\hat{I}}“)$ fixes a point of $K_{\hat{I}}“$. 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 $U^0(A_{\hat{I}}“)$.

30 pages, 6 figures. Minor edits. This is the final, accepted version, to appear in Algebraic and Geometric Topology