Convergence of normalized Betti numbers in nonpositive curvature

Abstract: I will show that if X is any symmetric space other than 3-dimensional hyperbolic space and M is any finite volume manifold that is a quotient of X, then the normalized Betti numbers of M are "testable", i.e. one can guess their values by sampling the manifold at random places. This is joint with Abert, Biringer and Bergeron, and extends some of our older work with Nikolov, Raimbault and Samet. The content of the recent paper involves a random discretization process that converts the "thick part" of M into a simplicial complex, together with analysis of the "thin parts" of M. As a corollary, we obtain that whenever X is a higher rank irreducible symmetric space and M_i is any sequence of distinct finite volume quotients of X, the normalized Betti numbers of the M_i converge to the "L^2-Betti numbers" of X.

differential geometrydynamical systemsgeometric topologysymplectic geometry

Audience: researchers in the topic

Geometry and Dynamics seminar

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