Spectrum and Growth on Infinite Covers

Richard Sharp (University of Warwick)

Abstract

It is easy to see that zero is an eigenvalue of the Laplacian on a compact manifold. If one takes an infinite cover of the manifold, is zero still in the spectrum? This question turns out to have a nice answer, known since the 1980s, which is related to the Banach—Tarski Paradox. We will discuss this and a related dynamical problem. In the latter case, we have a new result which describes what happens when a crucial symmetry assumption is dropped – this is joint work with Rhiannon Dougall.