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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
LERF and the Lubotzky–Sarnak Conjecture

Marc Lackenby, Darren D Long and Alan W Reid

Geometry & Topology 12 (2008) 2047–2056
Abstract

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.

Keywords
subgroup separability, Cheeger constant, Lubotzky–Sarnak conjecture
Mathematical Subject Classification 2000
Primary: 57M50
References
Publication
Received: 11 April 2008
Accepted: 21 May 2008
Published: 24 July 2008
Proposed: Walter Neumann
Seconded: Jean-Pierre Otal, Cameron Gordon
Authors
Marc Lackenby
Mathematical Institute
University of Oxford
24-29 St Giles’
Oxford OX1 3LB
UK
Darren D Long
Department of Mathematics
University of California
Santa Barbara
CA 93106
USA
Alan W Reid
Department of Mathematics
University of Texas
Austin
TX 78712
USA