# Quasi-actions on trees I. Bounded valence

### Abstract

Given a bounded valence, bushy tree $T$, we prove that any cobounded quasi-action of a group $G$ on $T$ is quasiconjugate to an action of $G$ on another bounded valence, bushy tree $T’$. This theorem has many applications: quasi-isometric rigidity for fundamental groups of finite, bushy graphs of coarse ${\mathop{\rm PD}\nolimits}(n)$ groups for each fixed $n$; a generalization to actions on Cantor sets of Sullivan’s theorem about uniformly quasiconformal actions on the 2-sphere; and a characterization of locally compact topological groups which contain a virtually free group as a cocompact lattice. Finally, we give the first examples of two finitely generated groups which are quasi-isometric and yet which cannot act on the same proper geodesic metric space, properly discontinuously and cocompactly by isometries.

## Authors

Lee Mosher

Department of Mathematics and Computer Science, Rutgers University, Newark, NJ 07102, United States

Michah Sageev

Mathematics Department, Technion, Israel Institute of Technology, Haifa, 32000, Israel

Kevin Whyte

Department of Mathematics, The University of Chicago, Chicago, IL 60637, United States