Summary: Geometry & Topology 12 (2008) 20472056 2047
LERF and the LubotzkySarnak Conjecture
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.
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
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 .