## Geometry & Topology

### Lacunary hyperbolic groups

#### 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 $C$ 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.

#### Article information

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

Dates
Revised: 9 April 2009
Accepted: 10 March 2009
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513800288

Digital Object Identifier
doi:10.2140/gt.2009.13.2051

Mathematical Reviews number (MathSciNet)
MR2507115

Zentralblatt MATH identifier
1243.20056

Subjects
Secondary: 20F69: Asymptotic properties of groups

#### Citation

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. https://projecteuclid.org/euclid.gt/1513800288

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