2000/10/31 by Lee Mosher, Michah Sageev, Kevin Whyte
Mathematics · #math.GR
published as Ann. of Math. (2), Vol. 158 (2003), no.1, 115--164 · 50 pages, published version
arxiv created 2004/02/23 · arxiv updated 2009/11/30
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 PD(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.