Summary: Geometry & Topology 12 (2008) 2047­2056 2047
LERF and the Lubotzky­Sarnak Conjecture
MARC LACKENBY
DARREN D LONG
ALAN W REID
We prove that every closed hyperbolic 3­manifold 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 Lubotzky­Sarnak conjecture.
57M50
1 Introduction
We begin by recalling the definition of Property . Let X be a finite graph, and let
V.X/ denote its vertex set. For any subset A of V.X/, let @A denote those edges
with one endpoint in A and one not in A. Define the Cheeger constant of X to be
h.X/ D min
j@Aj
jAj
W A V.X/ and 0 < jAj Ä jV.X/j=2 :
Now let G be a group with a finite symmetric generating set S . For any subgroup
Gi of G, let X.G=GiI S/ be the Schreier coset graph of G=Gi with respect to S .

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara
Reid, Alan - Department of Mathematics, University of Texas at Austin

Collections: Mathematics