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