Tree-graded spaces and asymptotic cones of groups

Details Tree-graded spaces and asymptotic cones of groups Tree-graded spaces and asymptotic cones of groups

2004-05-03

arxiv.org/abs/math/0405030v3

Mathematics

Geometric Topology

96 pages. The paper is accepted in "Topology". We revised the problem section adding a couple of problems. We introduced conce

Scientific paper

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.

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