Let $X$ be a compact Kahler surface which is a ball quotient. Can such $X$ contain a torus $T$ such that the fudamental class of $T$ is non-trivial? I expect this is false as $\pi\_{1}(X)$ is a hyperbolic group, thus $\mathbb{Z} \times \mathbb{Z}$ can not occur as a subgroup of $\pi\_{1}(X)$ (here I consider the subgroup generated by the loops of the torus $T$). I can also handle the case when the subgroup is $\mathbb{Z} \times \mathbb{Z}/n\mathbb{Z}$, but I am not able to exclude the cases when the subgroup is $0$, $\mathbb{Z}$, or $\mathbb{Z}/n\mathbb{Z}$ for $n \geq 2$. Can these cases occur? 



Question on Ball Quotients
Fundamental group of $X$ is torsion-free, so the image of $\eta: \pi\_1(T^2)\to \pi\_1(X)$ is either trivial or infinite cyclic. In any case, you can realize $\eta$ by a composition of maps
$$
T^2\to S^1\to X. 
$$
The first map will kill the fundamental class of the torus, since $H\_2(S^1)=0$. 


