The locally symmetric space SL(n,Z)\SL(n,R)/SO(n), or the space of flat n-tori of unit volume, has immersed, totally geodesic, flat tori of dimension (n-1). These tori are natural candidates for nontrivial homology cycles of manifold covers of SL(n,Z)\SL(n,R)/SO(n). In joint work with Grigori Avramidi, we show that some of these (n-1)-dim tori give nontrivial rational homology cycles in congruence covers of the locally symmetric space SL(n,Z) \SL(n,R)/SO(n). We also show that the dimension of the subspace of the (n-1)-homology group spanned by flat (n-1)-tori grows as one goes up in congruence covers. I will attempt to explain these in a general audience talk, which means, in this case, the prerequisite for this talk is basic linear algebra.

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