Proper Actions of Automorphism Groups of Free Products of Finite Groups

If G is a free product of finite groups, let ΣAut1(G) denote all (necessarily symmetric) automorphisms of G that do not permute factors in the free product. We show that a McCullough–Miller and Gutierrez–Krstic derived (also see Bogley–Krstic) space of pointed trees is an $\underline{E} \Sigma {\rm Aut}_1(G)$-space for these groups.