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

### Matt T. Clay

Allegheny College, Meadville, United States

## Abstract

For a finitely generated free group $F_{n}$, of rank at least 2, any finite subgroup of $Out(F_{n})$ can be realized as a group of automorphisms of a graph with fundamental group $F_{n}$. This result, known as $Out(F_{n})$ 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 $Out(F_{n})$ 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.

## Cite this article

Matt T. Clay, A fixed point theorem for deformation spaces of $G$-trees. Comment. Math. Helv. 82 (2007), no. 2, pp. 237–246

DOI 10.4171/CMH/91