Cheeger constants and L^2 Betti numbers

Lewis Bowen (Texas)

Abstract: The Cheeger constant of an infinite-volume manifold is the greatest lower bound of surface-area-to-volume ratios of its compact sub-manifolds. It is zero for Euclidean space but positive for hyperbolic space. It bounds the zero-th eigenvalue of the Laplace operator and, in the case of geometrically finite real hyperbolic manifolds, bounds the Hausdorff dimension of the limit set of the fundamental group. We ask: how small can the Cheeger constant be of a real hyperbolic n-manifold with no boundary and free fundamental groupI'll explain some partial results on this question by making use of L^2 Betti numbers.