# Commentarii Mathematici Helvetici

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**Volume 82, Issue 2, 2007, pp. 237–246**

**DOI: 10.4171/CMH/91**

Published online: 2007-06-30

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

^{[1]}(1) Allegheny College, Meadville, United States

For a finitely generated free group `F _{n}`, of rank at least 2, any
finite subgroup of Out(

`F`) can be realized as a group of automorphisms of a graph with fundamental group

_{n}`F`. This result, known as Out(

_{n}`F`) 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(

_{n}`F`) and the mapping class group respectively. For a nonnegative integer

_{n}`n`we define a class of groups (

`GVP`(

`n`)) and prove a similar statement for their outer automorphism groups.

*Keywords: *`G`-tree, deformation space, Out(`F _{n}`) realization, Nielsen realization

Clay Matt: A fixed point theorem for deformation spaces of `G`-trees. *Comment. Math. Helv.* 82 (2007), 237-246. doi: 10.4171/CMH/91