Geometry & Topology

LERF and the Lubotzky–Sarnak Conjecture

Marc Lackenby, Darren D Long, and Alan W Reid

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We prove that every closed hyperbolic 3–manifold has a family of (possibly infinite sheeted) coverings with the property that the Cheeger constants in the family tend to zero. This is used to show that, if in addition the fundamental group of the manifold is LERF, then it satisfies the Lubotzky–Sarnak conjecture.

Article information

Geom. Topol., Volume 12, Number 4 (2008), 2047-2056.

Received: 11 April 2008
Accepted: 21 May 2008
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M50: Geometric structures on low-dimensional manifolds

subgroup separability Cheeger constant Lubotzky–Sarnak conjecture


Lackenby, Marc; Long, Darren D; Reid, Alan W. LERF and the Lubotzky–Sarnak Conjecture. Geom. Topol. 12 (2008), no. 4, 2047--2056. doi:10.2140/gt.2008.12.2047.

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  • I Agol, Tameness of hyperbolic 3–manifolds
  • I Agol, D Groves, J Manning, Residual finiteness, QCERF and fillings of hyperbolic groups
  • I Agol, D D Long, A W Reid, The Bianchi groups are separable on geometrically finite subgroups, Ann. of Math. $(2)$ 153 (2001) 599–621
  • L Bowen, Free groups in lattices
  • R Brooks, The spectral geometry of a tower of coverings, J. Differential Geom. 23 (1986) 97–107
  • J Button, Largeness of LERF and 1–relator groups
  • D Calegari, D Gabai, Shrinkwrapping and the taming of hyperbolic 3–manifolds, J. Amer. Math. Soc. 19 (2006) 385–446
  • S Y Cheng, S T Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math. 28 (1975) 333–354
  • L Clozel, Démonstration de la conjecture $\tau$, Invent. Math. 151 (2003) 297–328
  • D Cooper, D D Long, A W Reid, Bundles and finite foliations, Invent. Math. 118 (1994) 255–283
  • K Corlette, Hausdorff dimensions of limit sets. I, Invent. Math. 102 (1990) 521–541
  • L Greenberg, Finiteness theorems for Fuchsian and Kleinian groups, from: “Discrete groups and automorphic functions (Proc. Conf., Cambridge, 1975)”, Academic Press, London (1977) 199–257
  • F Haglund, D T Wise, Special cube complexes, Geom. Funct. Anal. 17 (2008) 1551–1620
  • M Lackenby, Surface subgroups of Kleinian groups with torsion
  • M Lackenby, Heegaard splittings, the virtually Haken conjecture and property $(\tau)$, Invent. Math. 164 (2006) 317–359
  • M Lackenby, D D Long, A W Reid, Covering spaces of arithmetic $3$–orbifolds, to appear in International Math. Research Notices
  • D D Long, Engulfing and subgroup separability for hyperbolic groups, Trans. Amer. Math. Soc. 308 (1988) 849–859
  • D D Long, A W Reid, Surface subgroups and subgroup separability in 3-manifold topology, Publicações Matemáticas do IMPA, Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro (2005)
  • A Lubotzky, Discrete groups, expanding graphs and invariant measures, Progress in Mathematics 125, Birkhäuser Verlag, Basel (1994) With an appendix by Jonathan D Rogawski
  • A Lubotzky, Free quotients and the first Betti number of some hyperbolic manifolds, Transform. Groups 1 (1996) 71–82
  • A Lubotzky, A Zuk, On Property $\tau$, Monograph to appear
  • C T McMullen, Hausdorff dimension and conformal dynamics. I. Strong convergence of Kleinian groups, J. Differential Geom. 51 (1999) 471–515
  • P Scott, Subgroups of surface groups are almost geometric, J. London Math. Soc. $(2)$ 17 (1978) 555–565
  • D Sullivan, Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups, Acta Math. 153 (1984) 259–277
  • D Sullivan, Related aspects of positivity in Riemannian geometry, J. Differential Geom. 25 (1987) 327–351