Lacunary hyperbolic groups

By A. Ol’shanskii, Denis V. Osin, Mark V. Sapir. 2007

124 citations recorded by the source corpus.

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 1.C/DZ). 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 ! ; <

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e1b79c7f88a54c42906b4df904e163a1567478f3

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In Applied Mathematics

1. Wen-yuan Yang
2. Dominik Gruber
3. Michael Hull
4. Goulnara Arzhantseva
5. Christopher H. Cashen
6. David Hume
7. Denis V. Osin
8. Ashot Minasyan

In Geometry and Topology

1. Denis V. Osin
2. Wen-yuan Yang
3. Michael Hull
4. Dominik Gruber
5. Goulnara Arzhantseva
6. Christopher H. Cashen
7. David Hume
8. Hung Cong Tran

In Mathematical Physics

1. Denis V. Osin
2. Wen-yuan Yang
3. Michael Hull
4. Dominik Gruber
5. Goulnara Arzhantseva
6. Christopher H. Cashen
7. David Hume
8. Alessandro Sisto

Citations in the corpus

• Transitivity degrees of countable groups and acylindrical hyperbolicity (2015, background): “It is likely that lacunary hyperbolic amenable groups constructed in [46] are MIF.”
• Characterizations of Morse Quasi-Geodesics via Superlinear Divergence and Sublinear Contraction (2016, background): “There are groups that admit quasi-geodesics with superlinear divergence, yet have an asymptotic cone with no cut point, and therefore no Morse quasi-geodesics [26].”
• Statistically convex-cocompact actions of groups with contracting elements (2016, background): “ of groups with non-trivial Floyd boundary. In [41], V. Gerasimov proved that any non-elementary relatively hyperbolic groups have non-trivial Floyd boundary (the converse is an open problem posed in [66]). A. Karlsson [51] proved that a group with a non-trivial Floyd boundary is a convergence group”
• Divergence in lattices in semisimple Lie groups and graphs of groups (2008, unknown): “Note that there are finitely generated groups whose divergence is not linear but is arbitrarily close to being linear (and in fact is bounded by a linear function on arbitrary long intervals) [OOS05].”