Finite groups of untwisted outer automorphisms of RAAGs

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