Suppose we have a reductive group $G$ over $\mathbb{Q}$, a compact subgroup of the adelic points $K_f\subset G(\mathbb{A}_{\mathbb{Q}})$, and the associated locally symmetric space $$Y_K := G(\mathbb{Q} )\backslash G(\mathbb{A}_{\mathbb{Q}})/Z(\mathbb{A}_{\mathbb{Q}})K_fK_\infty.$$ The image of aany $\mathbb{Q}$-torus $T$ of rank $r$ (modulo the center) inside $G$ which is split at infinity should form a geodesic submanifold of dimension $r$ in $Y_K$. If $T$ is anisotropic (modulo the center), the submanifold is closed and hence represents a class in the homology $H_r(Y_K)$. In general, the image of an arbitrary torus split at infinity should represent a class in the cohomology $H_r(X_K, \text{cusps})$ for some suitable compactification.
Question: Can we describe the span of these classes? Are there cases where we can prove they generate that whole degree of homology, or close to it?
I'm especially interested in the case of maximal tori, in homological degree equal to the rank modulo center of $G$. In the $\text{GL}_2$ case, these generate all of homology (and are known classically as real quadratic geodesics and modular symbols, in the anisotropic resp. split cases.
