• Benjamini-Schramm convergence
• Cheeger constants
• hyperbolic manifolds
• L^2-Betti numbers
• metric measure spaces

The Cheeger constant of an infinite volume Riemannian n-manifold M is defined as the infimum, among smooth compact n-submanifolds K, of the ratio of the boundary area of K to its volume.

L^2-Betti numbers are an equivariant version of Betti numbers, defined for a group acting on a classifying space by using a homological theory that allows infinite square-summable chains.

Lewis Bowen in 2013 put together these two concepts, establishing a nontrivial lower bound for Cheeger constants of manifolds of the form X/G where X is a contractible Riemannian manifold and G is a discrete group belonging to a family defined using L^2-Betti numbers.

The proof involves different techniques, such as Benjamini-Schramm convergence of simplicial complexes and of metric measure spaces, and Lück's Approximation Theorem for L^2-Betti numbers.

Consequences of the result include a nontrivial lower bound for the Cheeger constant of H^(2n)/G where n>=2 and G is the fundamental group of a complete finite volume hyperbolic 3-manifold, a lower bound for the spectrum of the Laplace-Beltrami operator of such manifolds and an upper bound for the Hausdorff dimension of conical limit sets for the action of such a G on H^(2n). The result also implies that if n>=2 and G is a non-abelian free group then the bounded fundamental class of H^(2n)/G vanishes.