Geometry and Topology Seminar

Let M be a manifold. I'll discuss the notion of `strong convergence' of a sequence of finite dimensional unitary representations of the fundamental group of M. Once this convergence property is established for particular sequences of representations, it can be used to deduce information about the spectral gap of the Laplacian on covering spaces of M, or on vector bundles over M. This has led to several recent advances. For example, it is now known that every compact hyperbolic surface has a sequence of covering spaces with asymptotically optimal relative spectral gap. I'll discuss what is known and conjectured for higher dimensional hyperbolic manifolds.