Flat cycles in the homology of €$\Gamma$ \ SL${}_m\mathbb{R}$ / SO($m$)

Abstract

In this paper we show that flat ($m-1$)-dimensional tori give nontrivial rational homology cycles in congruence covers of the locally symmetric space SL${}_m\mathbb{Z}$ \ SL${}_m\mathbb{R}$ / SO($m$). We also show that the dimension of the subspace of $H_{m-1}(\Gamma$ \ SL${}_m\mathbb{R}$ / SO($m$); $\mathbb{Q})$ spanned by flat $(m-1)$-tori grows as one goes up in congruence covers.