For a finitely generated free group Fn, of rank at least 2, any finite subgroup of Out(Fn) can be realized as a group of automorphisms of a graph with fundamental group Fn. This result, known as Out(Fn) 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 Out(Fn) and the mapping class group respectively. For a nonnegative integer n we define a class of groups (GVP(n)) and prove a similar statement for their outer automorphism groups.
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Matt T. Clay, A fixed point theorem for deformation spaces of <var>G</var>-trees. Comment. Math. Helv. 82 (2007), no. 2, pp. 237–246DOI 10.4171/CMH/91