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 $\pi_{1}(X)$?
2 Answers
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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).
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$\begingroup$ Thank you very much for your answer. Do you know if the subgroups of the form $\mathbb{Z} \oplus \mathbb{Z}_{n}$ can occur, where $n \geq 2$? $\endgroup$– DavidDec 11, 2011 at 17:56
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$\begingroup$ Let me also ask a related question. Can $X$ as above contain a homologically essential real two dimensional submanifold that realizes the subgroup $\mathbb{Z} \oplus \mathbb{Z}_{n}$? I suspect the answer is no, but I am not able to prove it. $\endgroup$– DavidDec 11, 2011 at 18:22
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$\begingroup$ They can appear as subgroups of $PU(2,1)$, but not as a fundamental group, because the finite order elements have fixed points in the complex hyperbolic plane. $\endgroup$– BS.Dec 11, 2011 at 18:26
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I don't know what a ball quotient is, but the decisive value is the genus of $X$. If the genus is zero, $X$ is the projective space, if the genus is two, $X$ is a torus and the fundamental group is ${\mathbb Z}^2$ if the genus is $\ge 2$, then $X$ is a quotient of the upper half plane and the fundamental group $\pi$ is a uniform torsion-free lattice in $G=PSL_2({\mathbb R})$. Therefore every element is semisimple and the centralizer of any element $g\ne 1$ in $G$ is a torus, so the centralizer in $\pi$ is isomorphis to $\mathbb Z$. This means that in the case of genus $\ge 2$ the answer is $\mathbb Z$ only.
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4$\begingroup$ Anton, you are thinking of Riemann surfaces, which have complex dimension 1. The question was about complex surfaces, which have complex dimension 2. $\endgroup$– AngeloDec 11, 2011 at 9:12