Geodesic
Quasi-actions on trees I. Bounded valence
Lee Mosher1 ; Michah Sageev2 ; Kevin Whyte3
1 Department of Mathematics and Computer Science, Rutgers University, Newark, NJ 07102, United States
2 Mathematics Department, Technion, Israel Institute of Technology, Haifa, 32000, Israel
3 Department of Mathematics, The University of Chicago, Chicago, IL 60637, United States
Annals of mathematics, Tome 158 (2003) no. 1, pp. 115-164.

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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.
DOI : 10.4007/annals.2003.158.115

Lee Mosher 1 ; Michah Sageev 2 ; Kevin Whyte 3

1 Department of Mathematics and Computer Science, Rutgers University, Newark, NJ 07102, United States
2 Mathematics Department, Technion, Israel Institute of Technology, Haifa, 32000, Israel
3 Department of Mathematics, The University of Chicago, Chicago, IL 60637, United States 
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Lee Mosher; Michah Sageev; Kevin Whyte. Quasi-actions on trees I. Bounded valence. Annals of mathematics, Tome 158 (2003) no. 1, pp. 115-164. doi : 10.4007/annals.2003.158.115. http://geodesic.mathdoc.fr/articles/10.4007/annals.2003.158.115/

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