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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

Matt T. Clay[1]

(1) Allegheny College, Meadville, United States

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.

Keywords: G-tree, deformation space, Out(Fn) 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