LERF and the Lubotzky–Sarnak Conjecture

Lackenby, Marc; Long, Darren D; Reid, Alan W

doi:10.2140/gt.2008.12.2047

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LERF and the Lubotzky–Sarnak Conjecture

Marc Lackenby, Darren D Long and Alan W Reid

Geometry & Topology 12 (2008) 2047–2056

DOI: 10.2140/gt.2008.12.2047

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

Volume 12, issue 4 (2008)