  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.
