Convergence of normalized Betti numbers in nonpositive curvature

The normalized Betti numbers of a space M are the usual Betti numbers divided by the `volume’ of the space, which can mean the number of vertices if M is a simplicial complex, or the Riemannian volume if M is a Riemannian manifold. In 2009, Elek showed that normalized Betti numbers of bounded degree complexes M are `testable’, in the sense that they can be accurately guessed after analyzing the geometries of a finite number of bounded radius balls whose centers are chosen randomly in M. This testability is equivalent to a certain continuity statement: that when a sequence of such complexes `weakly converges’, the normalized Betti numbers converges. We’ll discuss to what extent the normalized Betti numbers of finite volume Riemannian manifolds M are testable. In particular, we will show that they are testable for quotients M of a fixed irreducible symmetric space X of noncompact type, as long as X is not H3.

This is joint work with Abert, Bergeron and Gelander.