Colloquium Sharp 4.2.15

Speaker: Richard Sharp (Warwick)

Title: Growth and spectra on regular covers

Abstract: Two natural numerical invariants that can be associated to a Riemannian manifold are the bottom of the spectrum of the Laplacian operator and, if the manifold are negatively curved, the exponential growth rate of closed geodesics. Suppose we have a regular cover of a compact manifold. Then, for each of these quantities, we might ask under what circumstances we have equality between the number associated to the cover and the number associated to the base.  This question becomes non-trivial questions once the cover is infinite. It turns out that the question has a common answer in the two cases and this depends only on the covering group as an abstract group. For the Laplacian, this result was obtained by Robert Brooks in the 1980s, and Rhiannon Dougall and I have recently obtained the analogue for the growth of closed geodesics. I will discuss this work, relating it to random walks and a class of groups introduced by von Neumann in his study of the Banach-Tarski Paradox.

Bio sketch: Richard Sharp is a mathematician at Warwick University, UK. He grew up in London and obtained a BSc in Mathematics from Imperial College, London, in 1987. This was followed by postgraduate study at Warwick, where he was supervised by William Parry, and he obtained his PhD, on the periodic orbit structure of hyperbolic flows, in 1990. After postdocs at IHES, Queen Mary, London, and Oxford, he moved to Manchester University in 1995, before returning to Warwick in 2012. He is mainly interested in ergodic theory and applications to geometry.