A fixed point theorem for deformation spaces of $G$-trees

Abstract

For a finitely generated free group $F_n$, of rank at least 2, any finite subgroup of $\mathrm{Out}(F_n)$ can be realized as a group of automorphisms of a graph with fundamental group $F_n$. This result, known as $\mathrm{Out}(F_n)$ realization, was proved by Zimmermann, Culler and Khramtsov. This theorem is comparable to Nielsen realization as proved by Kerckhoff: for a closed surface with negative Euler characteristic, any finite subgroup of the mapping class group can be realized as a group of isometries of a hyperbolic surface. Both of these theorems have restatements in terms of fixed points of actions on spaces naturally associated to $\mathrm{Out}(F_n)$ and the mapping class group respectively. For a nonnegative integer $n$ we define a class of groups $(\text{GVP}(n))$ and prove a similar statement for their outer automorphism groups.