LERF and the Lubotzky-Sarnak conjecture

TL;DR

This paper demonstrates that closed hyperbolic 3-manifolds have coverings with Cheeger constants approaching zero, and if their fundamental group is LERF, they satisfy the Lubotzky-Sarnak conjecture.

Contribution

It establishes a connection between LERF property of fundamental groups and the Lubotzky-Sarnak conjecture for hyperbolic 3-manifolds.

Findings

Existence of coverings with Cheeger constants tending to zero

LERF property implies satisfaction of the Lubotzky-Sarnak conjecture

Provides new insights into the structure of hyperbolic 3-manifolds

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.