### 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 π_{1}(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 language | English (US) |
---|---|

Pages (from-to) | 2051-2140 |

Number of pages | 90 |

Journal | Geometry and Topology |

Volume | 13 |

Issue number | 4 |

DOIs | |

State | Published - 2009 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*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.

Research output: Contribution to journal › Article

*Geometry and Topology*, vol. 13, no. 4, pp. 2051-2140. https://doi.org/10.2140/gt.2009.13.2051

}

TY - JOUR

T1 - Lacunary hyperbolic groups

AU - Ol'shanskii, Alexander Y.

AU - Osin, Denis V.

AU - Sapir, Mark V.

AU - Kapovich, Michael

AU - Kleiner, Bruce

PY - 2009

Y1 - 2009

N2 - 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.

AB - 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.

KW - Asymptotic cone

KW - Cut point

KW - Directed limit

KW - Fundamental group

KW - Hyperbolic group

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UR - http://www.scopus.com/inward/citedby.url?scp=74249083363&partnerID=8YFLogxK

U2 - 10.2140/gt.2009.13.2051

DO - 10.2140/gt.2009.13.2051

M3 - Article

AN - SCOPUS:74249083363

VL - 13

SP - 2051

EP - 2140

JO - Geometry and Topology

JF - Geometry and Topology

SN - 1364-0380

IS - 4

ER -