Geometric characterization of the Q-rank by closed flats

Let $\widetilde{X}$ be a symmetric space with no Euclidean local factors, let $X=\Gamma\backslash\widetilde{X}$ be a finite-volume locally symmetric space, and let $\operatorname{\mathbb{Q}\text{-}rank}\Gamma$ denote its algebraic $\mathbb{Q}$-rank. A flat is a connected, totally geodesic, flat submanifold, and a flat is topologically closed and simply connected when it has those properties in $X$. Geometric characterization of the Q-rank. The number $\operatorname{\mathbb{Q}\text{-}rank}\Gamma$ is the largest natural number $r$ such that some finite cover of $X$ contains a topologically closed, simply connected, $r$-dimensional flat. The preceding text presents this as a geometric interpretation that would follow if the divergent torus orbit conjecture holds. Since B. Weiss is stated to have proved that conjecture in all cases, this characterization is solved as well.