Quasi-actions on trees I. Bounded valence

Details Quasi-actions on trees I. Bounded valence Quasi-actions on trees I. Bounded valence

2000-10-13

arxiv.org/abs/math/0010136v2

Ann. of Math. (2), Vol. 158 (2003), no.1, 115--164

Mathematics

Group Theory

50 pages, published version

Scientific paper

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.

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