Tree-graded spaces and asymptotic cones of groups

• Published in: Topology (Oxford) Vol. 44; no. 5; pp. 959 - 1058
• Main Authors: Druţu, Cornelia, Sapir, Mark
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 2005; Elsevier
• ISSN: 0040-9383; 1879-3215
• DOI: 10.1016/j.top.2005.03.003

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 a finitely generated group with a continuum of non- π 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.