On the growth of Betti numbers of locally symmetric spaces
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 Lück 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.
 Publication:

arXiv eprints
 Pub Date:
 April 2011
 arXiv:
 arXiv:1104.5559
 Bibcode:
 2011arXiv1104.5559A
 Keywords:

 Mathematics  Group Theory;
 Mathematics  Geometric Topology
 EPrint:
 Announcement