On automorphism groups of free products of finite groups, I: Proper Actions

If $G$ is a free product of finite groups, let $Sigma Aut_1(G)$ denote all (necessarily symmetric) automorphisms of $G$ that do not permute factors in the free product. We show that a McCullough-Miller [D. McCullough and A. Miller, {em Symmetric Automorphisms of Free Products}, Mem. Amer. Math. Soc. 122 (1996), no. 582] and Guti{e}rrez-Krsti{c} [M. Guti{e}rrez and S. Krsti{c}, {em Normal forms for the group of basis-conjugating automorphisms of a free group}, International Journal of Algebra and Computation 8 (1998) 631-669] derived (also see Bogley-Krsti{c} [W. Bogley and S. Krsti{c}, {em String groups and other subgroups of $Aut(F_n)$}, preprint] space of pointed trees is an $underline{E} Sigma Aut_1(G)$-space for these groups.