Geometry & Topology

Free groups in lattices

Lewis Bowen

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Let G be any locally compact unimodular metrizable group. The main result of this paper, roughly stated, is that if F<G is any finitely generated free group and Γ<G any lattice, then up to a small perturbation and passing to a finite index subgroup, F is a subgroup of Γ. If GΓ is noncompact then we require additional hypotheses that include G= SO(n,1).

Article information

Geom. Topol., Volume 13, Number 5 (2009), 3021-3054.

Received: 27 May 2007
Revised: 26 August 2009
Accepted: 17 August 2009
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20E07: Subgroup theorems; subgroup growth
Secondary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 20F67: Hyperbolic groups and nonpositively curved groups 22D40: Ergodic theory on groups [See also 28Dxx] 20E05: Free nonabelian groups

free group surface group Kleinian group limit set


Bowen, Lewis. Free groups in lattices. Geom. Topol. 13 (2009), no. 5, 3021--3054. doi:10.2140/gt.2009.13.3021.

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  • L Bowen, Periodicity and circle packings of the hyperbolic plane, Geom. Dedicata 102 (2003) 213–236
  • L Bowen, C Holton, C Radin, L Sadun, Uniqueness and symmetry in problems of optimally dense packings, Math. Phys. Electron. J. 11 (2005) Paper 1, 34 pp.
  • M R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grund. der Math. Wissenschaften 319, Springer, Berlin (1999)
  • M Coornaert, Mesures de Patterson–Sullivan sur le bord d'un espace hyperbolique au sens de Gromov, Pacific J. Math. 159 (1993) 241–270
  • M Coornaert, T Delzant, A Papadopoulos, Géométrie et théorie des groupes. Les groupes hyperboliques de Gromov, Lecture Notes in Math. 1441, Springer, Berlin (1990) With an English summary
  • A Dranishnikov, V Schroeder, Aperiodic colorings and tilings of Coxeter groups, Groups Geom. Dyn. 1 (2007) 311–328
  • V A Efremovič, The proximity geometry of Riemannian manifolds, Uspehi Matem. Nauk (N.S.) 8 (1953) 189–195
  • C Goodman-Strauss, A strongly aperiodic set of tiles in the hyperbolic plane, Invent. Math. 159 (2005) 119–132
  • M Gromov, Hyperbolic groups, from: “Essays in group theory”, Math. Sci. Res. Inst. Publ. 8, Springer, New York (1987) 75–263
  • M Lackenby, Surface subgroups of Kleinian groups with torsion, to appear in Invent. Math.
  • M Lackenby, D D Long, A W Reid, LERF and the Lubotzky–Sarnak conjecture, Geom. Topol. 12 (2008) 2047–2056
  • G A Margulis, S Mozes, Aperiodic tilings of the hyperbolic plane by convex polygons, Israel J. Math. 107 (1998) 319–325
  • J D Masters, Kleinian groups with ubiquitous surface subgroups, Groups Geom. Dyn. 2 (2008) 263–269
  • C T McMullen, Hausdorff dimension and conformal dynamics. I. Strong convergence of Kleinian groups, J. Differential Geom. 51 (1999) 471–515
  • J Milnor, A note on curvature and fundamental group, J. Differential Geometry 2 (1968) 1–7
  • R Penrose, Pentaplexity: a class of nonperiodic tilings of the plane, Eureka 39 (1978) 16–32 Reproduced in Math. Intell. 2 (1979) 32–37
  • A S Švarc, A volume invariant of coverings, Dokl. Akad. Nauk SSSR $($N.S.$)$ 105 (1955) 32–34
  • W A Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math. 97 (1989) 553–583