Let $X$ be a compact complex surface of general type which a ball quotient. Is it true that $\pi\_{1}(X)$ can not contain ${\mathbb{Z}}^{2}$ as a subgroup? What kind of infinite abelian groups can occur as a subgroup of $\pi\_{1}(X)$?



Abelian subgroups of ball quotient
If I understand your question, you ask if a cocompact torsion free subgroup of $PU(2,1)$ (namely $\Gamma=\pi\_{1}(X)$) can contain a ${\mathbb Z}^2$. 


This is not the case, because $\Gamma$ is a Gromov-hyperbolic group <http://en.wikipedia.org/wiki/Hyperbolic_group>, due to the fact that $X$ has a negatively curved riemannian metric (quotient of that of the complex hyperbolic plane, aka 4-ball).


