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Lacunary hyperbolic groups

Alexander Yu Ol'shanskii, Denis V Osin and Mark V Sapir

Appendix: Michael Kapovich and Bruce Kleiner

Geometry & Topology 13 (2009) 2051–2140

DOI: 10.2140/gt.2009.13.2051


We call a finitely generated group lacunary hyperbolic if one of its asymptotic cones is an R–tree. We characterize lacunary hyperbolic groups as direct limits of Gromov hyperbolic groups satisfying certain restrictions on the hyperbolicity constants and injectivity radii. Using central extensions of lacunary hyperbolic groups, we solve a problem of Gromov by constructing a group whose asymptotic cone C has countable but nontrivial fundamental group (in fact C is homeomorphic to the direct product of a tree and a circle, so π1(C) = Z). We show that the class of lacunary hyperbolic groups contains non–virtually cyclic elementary amenable groups, groups with all proper subgroups cyclic (Tarski monsters) and torsion groups. We show that Tarski monsters and torsion groups can have so-called graded small cancellation presentations, in which case we prove that all their asymptotic cones are hyperbolic and locally isometric to trees. This allows us to solve two problems of Druţu and Sapir and a problem of Kleiner about groups with cut points in their asymptotic cones. We also construct a finitely generated group whose divergence function is not linear but is arbitrarily close to being linear. This answers a question of Behrstock.


hyperbolic group, directed limit, asymptotic cone, cut point, fundamental group

Mathematical Subject Classification

Primary: 20F65

Secondary: 20F69


Received: 17 July 2007
Revised: 9 April 2009
Accepted: 10 March 2009
Published: 30 April 2009
Proposed: Benson Farb
Seconded: Dmitri Burago, Walter Neumann

Alexander Yu Ol'shanskii
Department of Mathematics
Vanderbilt University
Nashville, TN 37240
Department of Mathematics
Moscow State University
Moscow 119899
Denis V Osin
Department of Mathematics
Vanderbilt University
Nashville, TN 37240
Mark V Sapir
Department of Mathematics
Vanderbilt University
Nashville, TN 37240
Michael Kapovich
Department of Mathematics
University of California
Davis, CA 95616
Bruce Kleiner
Department of Mathematics
Courant Institute
New York University
251 Mercer Street
New York, NY 10012