On rigidity of group actions on trees
by
Max Forester
University of Warwick

Let G be a group. A G-tree is a simplicial tree with a G-action, without inversions. It corresponds to a graph of groups decomposition of G. We consider conditions under which a G-tree in uniquely determined by G alone. Our method is to study deformations of G-trees. There are several equivalent conditions characterizing when two G-trees are related by a deformation. We then show that if a G-tree X is ``strongly slide-free, '' then it is the unique such tree among all trees deformation-equivalent to X. These methods allow us to extend the rigidity theorem of Bass and Lubotzky to G-trees that are not locally finite. We also apply the results to generalized Baumslag-Solitar groups, yielding canonical decompositions of certain of these groups. We also show that strongly slide-free G-trees are quasi-isometrically rigid, in an equiariant sense.

Date received: July 15, 2001