Volume 13, issue 4 (2009)

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Lacunary hyperbolic groups

Appendix: Michael Kapovich and Bruce Kleiner

Geometry & Topology 13 (2009) 2051–2140
Abstract

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 $\mathsc{C}$ has countable but nontrivial fundamental group (in fact $\mathsc{C}$ is homeomorphic to the direct product of a tree and a circle, so ${\pi }_{1}\left(\mathsc{C}\right)=ℤ$). 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.

Keywords
hyperbolic group, directed limit, asymptotic cone, cut point, fundamental group
Primary: 20F65
Secondary: 20F69
Publication
Received: 17 July 2007
Revised: 9 April 2009
Accepted: 10 March 2009
Published: 30 April 2009
Proposed: Benson Farb
Seconded: Dmitri Burago, Walter Neumann
Authors
 Alexander Yu Ol’shanskii Department of Mathematics Vanderbilt University Nashville, TN 37240 USA Department of Mathematics Moscow State University Moscow 119899 Russia Denis V Osin Department of Mathematics Vanderbilt University Nashville, TN 37240 USA Mark V Sapir Department of Mathematics Vanderbilt University Nashville, TN 37240 USA Michael Kapovich Department of Mathematics University of California Davis, CA 95616 USA Bruce Kleiner Department of Mathematics Courant Institute New York University 251 Mercer Street New York, NY 10012 USA