Counting flat cycles in the homology of locally symmetric spaces

Locally symmetric spaces like $SL(n,\mathbb{Z})\mathrm{\backslash }S{L}_n(\mathbb{R})\mathrm{/}SO(n)$ contain immersed compact flat manifolds of dimension equal to the real rank. We give a lower bound for the contribution of these cycles to the homology of congruence covers. Similar results are proved for other families of locally symmetric spaces.