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.
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