LERF and the Lubotzky-Sarnak Conjecture M. Lackenby Summary: LERF and the Lubotzky-Sarnak Conjecture M. Lackenby , D. D. Long & A. W. Reid April 10, 2008 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) = min |A| |A| : A V (X) and 0 < |A| |V (X)|/2 . Now let G be a group with a finite symmetric generating set S. For any subgroup Gi of G, let X(G/Gi; S) be the Schreier coset graph of G/Gi with respect to S. Then G is said to have Property with respect to a collection of finite index subgroups {Gi} if infi h(X(G/Gi; S)) > 0. This turns out not to depend on the choice of finite generating set S. Also, G is said to have Property if it has Property with respect to the collection of all subgroups of finite index in G. In the context of of finite volume hyperbolic manifolds, Lubotzky and Sarnak made the following conjecture. Collections: Mathematics