Geometry & Topology

Lacunary hyperbolic groups

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

Full-text: Open access

Abstract

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

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

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

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

Keywords
hyperbolic group directed limit asymptotic cone cut point fundamental group

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


Export citation

References

  • S I Adian, The Burnside problem and identities in groups, Ergebnisse der Math. und ihrer Grenzgebiete 95, Springer, Berlin (1979) Translated from the Russian by J Lennox and J Wiegold
  • J M Alonso, T Brady, D Cooper, V Ferlini, M Lustig, M Mihalik, H Short (editor), Notes on word hyperbolic groups, from: “Group theory from a geometrical viewpoint (Trieste, 1990)”, (E Ghys, A Haefliger, A Verjovsky, editors), World Sci. Publ., River Edge, NJ (1991) 3–63
  • G Arzhantseva, A Minasyan, D V Osin, The SQ–universality and residual properties of relatively hyperbolic groups, J. Algebra 315 (2007) 165–177
  • W Ballmann, Lectures on spaces of nonpositive curvature, DMV Seminar 25, Birkhäuser Verlag, Basel (1995) With an appendix by M Brin
  • J A Behrstock, Asymptotic geometry of the mapping class group and Teichmüller space, Geom. Topol. 10 (2006) 1523–1578
  • B H Bowditch, Notes on Gromov's hyperbolicity criterion for path-metric spaces, from: “Group theory from a geometrical viewpoint (Trieste, 1990)”, (E Ghys, A Haefliger, A Verjovsky, editors), World Sci. Publ., River Edge, NJ (1991) 64–167
  • M R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grund. der Math. Wissenschaften 319, Springer, Berlin (1999)
  • J Burillo, Dimension and fundamental groups of asymptotic cones, J. London Math. Soc. $(2)$ 59 (1999) 557–572
  • L van den Dries, A J Wilkie, Gromov's theorem on groups of polynomial growth and elementary logic, J. Algebra 89 (1984) 349–374
  • C Druţu, Relatively hyperbolic groups: geometry and quasi-isometric invariance, to appear in Comm. Math. Helv.
  • C Druţu, Quasi-isometry invariants and asymptotic cones, from: “International Conference on Geometric and Combinatorial Methods in Group Theory and Semigroup Theory (Lincoln, NE, 2000)”, Internat. J. Algebra Comput. 12 (2002) 99–135
  • C Druţu, S Mozes, M V Sapir, Divergence in lattices in semisimple Lie groups and graphs of groups, to appear in Trans. Amer. Math. Soc.
  • C Druţu, M V Sapir, Tree-graded spaces and asymptotic cones of groups, Topology 44 (2005) 959–1058 With an appendix by D V Osin and Sapir
  • C Druţu, M V Sapir, Groups acting on tree-graded spaces and splittings of relatively hyperbolic groups, Adv. Math. 217 (2008) 1313–1367
  • A Erschler, D V Osin, Fundamental groups of asymptotic cones, Topology 44 (2005) 827–843
  • W J Floyd, Group completions and limit sets of Kleinian groups, Invent. Math. 57 (1980) 205–218
  • S M Gersten, Bounded cocycles and combings of groups, Internat. J. Algebra Comput. 2 (1992) 307–326
  • É Ghys, P de la Harpe, Espaces métriques hyperboliques, from: “Sur les groupes hyperboliques d'après Mikhael Gromov (Bern, 1988)”, Progr. Math. 83, Birkhäuser Boston, Boston, MA (1990) 27–45
  • M Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. (1981) 53–73
  • M Gromov, Hyperbolic groups, from: “Essays in group theory”, (S Gersten, editor), Math. Sci. Res. Inst. Publ. 8, Springer, New York (1987) 75–263
  • M Gromov, Asymptotic invariants of infinite groups, from: “Geometric group theory, Vol. 2 (Sussex, 1991)”, (G A Niblo, M A Roller, editors), London Math. Soc. Lecture Note Ser. 182, Cambridge Univ. Press (1993) 1–295
  • M Gromov, Random walk in random groups, Geom. Funct. Anal. 13 (2003) 73–146
  • P Hall, The Edmonton notes on nilpotent groups, Queen Mary College Math. Notes, Math. Dept., Queen Mary College, London (1969)
  • G Higman, Amalgams of $p$–groups, J. Algebra 1 (1964) 301–305
  • S V Ivanov, The free Burnside groups of sufficiently large exponents, Internat. J. Algebra Comput. 4 (1994) ii+308
  • S V Ivanov, A Y Ol'shanskii, Hyperbolic groups and their quotients of bounded exponents, Trans. Amer. Math. Soc. 348 (1996) 2091–2138
  • M Kapovich, B Kleiner, B Leeb, Quasi-isometries and the de Rham decomposition, Topology 37 (1998) 1193–1211
  • M Kapovich, B Leeb, $3$–manifold groups and nonpositive curvature, Geom. Funct. Anal. 8 (1998) 841–852
  • A Karlsson, Free subgroups of groups with nontrivial Floyd boundary, Comm. Algebra 31 (2003) 5361–5376
  • B Kleiner, B Leeb, Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings, Inst. Hautes Études Sci. Publ. Math. (1997) 115–197 (1998)
  • M Koubi, Croissance uniforme dans les groupes hyperboliques, Ann. Inst. Fourier $($Grenoble$)$ 48 (1998) 1441–1453
  • L Kramer, S Shelah, K Tent, S Thomas, Asymptotic cones of finitely presented groups, Adv. Math. 193 (2005) 142–173
  • R C Lyndon, P E Schupp, Combinatorial group theory, Ergebnisse der Math. und ihrer Grenzgebiete 89, Springer, Berlin (1977)
  • A I Mal'cev, Model correspondences, Izv. Akad. Nauk SSSR. Ser. Mat. 23 (1959) 313–336
  • K V Mikhajlovskii, A Y Ol'shanskii, Some constructions relating to hyperbolic groups, from: “Geometry and cohomology in group theory (Durham, 1994)”, (P H Kropholler, G A Niblo, R St öhr, editors), London Math. Soc. Lecture Note Ser. 252, Cambridge Univ. Press (1998) 263–290
  • I Mineyev, Straightening and bounded cohomology of hyperbolic groups, Geom. Funct. Anal. 11 (2001) 807–839
  • W D Neumann, L Reeves, Central extensions of word hyperbolic groups, Ann. of Math. $(2)$ 145 (1997) 183–192
  • Y Ollivier, Growth exponent of generic groups, Comment. Math. Helv. 81 (2006) 569–593
  • A Y Ol'shanskii, An infinite simple torsion-free Noetherian group, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979) 1328–1393
  • A Y Ol'shanskii, An infinite group with subgroups of prime orders, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980) 309–321, 479
  • A Y Ol'shanskii, Geometry of defining relations in groups, Math. and its Applications (Soviet Ser.) 70, Kluwer, Dordrecht (1991) Translated from the 1989 Russian original by Yu A Bakhturin
  • A Y Ol'shanskii, On residualing homomorphisms and $G$–subgroups of hyperbolic groups, Internat. J. Algebra Comput. 3 (1993) 365–409
  • A Y Ol'shanskii, M V Sapir, Groups with small Dehn functions and bipartite chord diagrams, Geom. Funct. Anal. 16 (2006) 1324–1376
  • A Y Ol'shanskii, M V Sapir, A finitely presented group with two non-homeomorphic asymptotic cones, Internat. J. Algebra Comput. 17 (2007) 421–426
  • D V Osin, Kazhdan constants of hyperbolic groups, Funktsional. Anal. i Prilozhen. 36 (2002) 46–54
  • D V Osin, Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems, Mem. Amer. Math. Soc. 179 (2006) vi+100
  • P Papasoglu, On the asymptotic cone of groups satisfying a quadratic isoperimetric inequality, J. Differential Geom. 44 (1996) 789–806
  • F Point, Groups of polynomial growth and their associated metric spaces, J. Algebra 175 (1995) 105–121
  • J Rosenberg, $C\sp{\ast} $–algebras, positive scalar curvature, and the Novikov conjecture, Inst. Hautes Études Sci. Publ. Math. (1983) 197–212 (1984)
  • G Skandalis, J L Tu, G Yu, The coarse Baum–Connes conjecture and groupoids, Topology 41 (2002) 807–834
  • S Thomas, B Velickovic, Asymptotic cones of finitely generated groups, Bull. London Math. Soc. 32 (2000) 203–208
  • A Yaman, A topological characterisation of relatively hyperbolic groups, J. Reine Angew. Math. 566 (2004) 41–89