On the growth of Betti numbers of locally symmetric spaces

TL;DR

This paper establishes a strong uniform version of the Lück Approximation Theorem for higher rank locally symmetric spaces, demonstrating the asymptotic behavior of Betti numbers as their volumes grow infinitely large.

Contribution

It introduces a novel adaptation of local convergence theory to Riemannian manifolds, strengthening the understanding of Betti number growth in large-volume locally symmetric spaces.

Findings

Betti numbers normalized converge as volume tends to infinity

Provides a uniform version of Lück Approximation Theorem

Shows local convergence of manifolds to universal cover

Abstract

We announce new results concerning the asymptotic behavior of the Betti numbers of higher rank locally symmetric spaces as their volumes tend to infinity. Our main theorem is a uniform version of the L\"uck Approximation Theorem \cite{luck}, which is much stronger than the linear upper bounds on Betti numbers given by Gromov in \cite{BGS}. The basic idea is to adapt the theory of local convergence, originally introduced for sequences of graphs of bounded degree by Benjamimi and Schramm, to sequences of Riemannian manifolds. Using rigidity theory we are able to show that when the volume tends to infinity, the manifolds locally converge to the universal cover in a sufficiently strong manner that allows us to derive the convergence of the normalized Betti numbers.