Geometry & Topology

Lacunary hyperbolic groups

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

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We call a finitely generated group lacunary hyperbolic if one of its asymptotic cones is an –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)=). 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.

Article information

Geom. Topol., Volume 13, Number 4 (2009), 2051-2140.

Received: 17 July 2007
Revised: 9 April 2009
Accepted: 10 March 2009
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]
Secondary: 20F69: Asymptotic properties of groups

hyperbolic group directed limit asymptotic cone cut point fundamental group


Ol’shanskii, Alexander Yu; Osin, Denis V; Sapir, Mark V. Lacunary hyperbolic groups. Geom. Topol. 13 (2009), no. 4, 2051--2140. doi:10.2140/gt.2009.13.2051.

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