Deformation and rigidity of simplicial group actions on trees

Abstract

We study a notion of deformation for simplicial trees with group actions ($G$–trees). Here $G$ is a fixed, arbitrary group. Two $G$–trees are related by a deformation if there is a finite sequence of collapse and expansion moves joining them. We show that this relation on the set of $G$–trees has several characterizations, in terms of dynamics, coarse geometry, and length functions. Next we study the deformation space of a fixed $G$–tree $X$. We show that if $X$ is “strongly slide-free” then it is the unique reduced tree in its deformation space. These methods allow us to extend the rigidity theorem of Bass and Lubotzky to trees that are not locally finite. This yields a unique factorization theorem for certain graphs of groups. We apply the theory to generalized Baumslag–Solitar groups and show that many have canonical decompositions. We also prove a quasi-isometric rigidity theorem for strongly slide-free $G$–trees.