We prove that every closed hyperbolic –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.
"LERF and the Lubotzky–Sarnak Conjecture." Geom. Topol. 12 (4) 2047 - 2056, 2008. https://doi.org/10.2140/gt.2008.12.2047