TAILIEUCHUNG - Đề tài " Quasi-actions on trees I. Bounded valence "

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. . | Annals of Mathematics Quasi-actions on trees I. Bounded valence By Lee Mosher Michah Sageev and Kevin Whyte Annals of Mathematics 158 2003 115 164 Quasi-actions on trees I. Bounded valence By Lee Mosher Michah Sageev and Kevin WHyTE 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 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. 1. Introduction A quasi -action of a group G on a metric space X associates to each g G G a quasi-isometry Ag x g x of X with uniform quasi-isometry constants so that Aid Idx and so that the distance between Ag o Ah and Agh in the sup norm is uniformly bounded independent of g h G G. Quasi-actions arise naturally in geometric group theory if a metric space X is quasi-isometric to a finitely generated group G with its word metric then the left action of G on itself can be quasiconjugated to give a quasiaction of G on X . Moreover a quasi-action which arises in this manner is cobounded and proper these properties are generalizations of cocompact and properly discontinuous as applied to isometric actions. Given a metric space X a fundamental problem in geometric group theory is to characterize groups quasi-isometric to X or equivalently to characterize groups which have a proper cobounded quasi-action on X. A more general problem is to characterize arbitrary .

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