Lacunary hyperbolic groups

Alexander Y. Ol'shanskii, Denis V. Osin, Mark V. Sapir, Michael Kapovich, Bruce Kleiner

Research output: Contribution to journalArticle

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 socalled 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.

Original languageEnglish (US)
Pages (from-to)2051-2140
Number of pages90
JournalGeometry and Topology
Volume13
Issue number4
DOIs
StatePublished - 2009

Fingerprint

Hyperbolic Groups
Asymptotic Cone
Finitely Generated Group
Torsion
Direct Limit
R-tree
Amenable Group
Central Extension
Injectivity
Hyperbolicity
Direct Product
Homeomorphic
Cancellation
Fundamental Group
Isometric
Countable
Divergence
Circle
Radius
Subgroup

Keywords

  • Asymptotic cone
  • Cut point
  • Directed limit
  • Fundamental group
  • Hyperbolic group

ASJC Scopus subject areas

  • Geometry and Topology

Cite this

Ol'shanskii, A. Y., Osin, D. V., Sapir, M. V., Kapovich, M., & Kleiner, B. (2009). Lacunary hyperbolic groups. Geometry and Topology, 13(4), 2051-2140. https://doi.org/10.2140/gt.2009.13.2051

Lacunary hyperbolic groups. / Ol'shanskii, Alexander Y.; Osin, Denis V.; Sapir, Mark V.; Kapovich, Michael; Kleiner, Bruce.

In: Geometry and Topology, Vol. 13, No. 4, 2009, p. 2051-2140.

Research output: Contribution to journalArticle

Ol'shanskii, AY, Osin, DV, Sapir, MV, Kapovich, M & Kleiner, B 2009, 'Lacunary hyperbolic groups', Geometry and Topology, vol. 13, no. 4, pp. 2051-2140. https://doi.org/10.2140/gt.2009.13.2051
Ol'shanskii AY, Osin DV, Sapir MV, Kapovich M, Kleiner B. Lacunary hyperbolic groups. Geometry and Topology. 2009;13(4):2051-2140. https://doi.org/10.2140/gt.2009.13.2051
Ol'shanskii, Alexander Y. ; Osin, Denis V. ; Sapir, Mark V. ; Kapovich, Michael ; Kleiner, Bruce. / Lacunary hyperbolic groups. In: Geometry and Topology. 2009 ; Vol. 13, No. 4. pp. 2051-2140.
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