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

Abstract

If $G$ is a free product of finite groups, let $ΣAut_1(G)$ 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 Gutiérrez-Krstić [M. Gutiérrez and S. Krstić, {\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-Krstić [W. Bogley and S. Krstić, {\em String groups and other subgroups of $Aut(F_n)$}, preprint] space of pointed trees is an $\underline{E} ΣAut_1(G)$-space for these groups.