Cornelia Drutu, Mark Sapir
Published 2004-05-03, updated 2007-03-21Version 3
We introduce a concept of tree-graded metric space and we use it to show quasi-isometry invariance of certain classes of relatively hyperbolic groups, to obtain a characterization of relatively hyperbolic groups in terms of their asymptotic cones, to find geometric properties of Cayley graphs of relatively hyperbolic groups, and to construct the first example of finitely generated group with a continuum of non-$\pi_1$-equivalent asymptotic cones. Note that by a result of Kramer, Shelah, Tent and Thomas, continuum is the maximal possible number of different asymptotic cones of a finitely generated group, provided that the Continuum Hypothesis is true.
Comments: 96 pages. The paper is accepted in "Topology". We revised the problem section adding a couple of problems. We introduced concepts of constricted (unconstricted, wide) groups and slow asymptotic cones. The Morse lemma for relatively hyperbolic groups is improved thanks to a question from Chris Hruska. A result about asymptotic cones of uniformly amenable groups and a result about groups whose asymptotic cone is a real lines are added. We also revised the text accordig to the comments of the referee and other readers of the paper
Counting problems in graph products and relatively hyperbolic groups
Notions of Anosov representation of relatively hyperbolic groups
Complex of Relatively Hyperbolic Groups
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