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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? If yes, please let me know if you are aware of such examples.

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?

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 $\mathbb{Z}$ or $\mathbb{Z} \times \mathbb{Z}/n\mathbb{Z}$, but I am not able to exclude the cases when the subgroup is $0$, or $\mathbb{Z}/n\mathbb{Z}$ for $n \geq 2$. Can these cases occur? If yes, please let me know if you are aware of such examples.

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? If yes, please let me know if you are aware of such examples.

Let $X$ be a 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 and $\mathbb{Z} \times \mathbb{Z}$ can not occur as a subgroup (here I consider the subgroup given by the generators of the torus $T$). I can also handle the case when the subgroup is $\mathbb{Z}$, but I am not able to exclude the cases when the subgroup is $0$, $\mathbb{Z} \times \mathbb{Z}/n$, or $\mathbb{Z}/n\mathbb{Z}$ for $n \geq 2$. Can these cases occur? If yes, please let me know if you are aware of such examples.

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 $\mathbb{Z}$ or $\mathbb{Z} \times \mathbb{Z}/n\mathbb{Z}$, but I am not able to exclude the cases when the subgroup is $0$, or $\mathbb{Z}/n\mathbb{Z}$ for $n \geq 2$. Can these cases occur? If yes, please let me know if you are aware of such examples.