Abelian subgroups of ball quotient - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T07:17:12Z http://mathoverflow.net/feeds/question/83167 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/83167/abelian-subgroups-of-ball-quotient Abelian subgroups of ball quotient David 2011-12-11T07:38:12Z 2011-12-11T10:32:35Z <p>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 $\pi_{1}(X)$?</p> http://mathoverflow.net/questions/83167/abelian-subgroups-of-ball-quotient/83170#83170 Answer by anton for Abelian subgroups of ball quotient anton 2011-12-11T09:06:05Z 2011-12-11T09:06:05Z <p>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.</p> http://mathoverflow.net/questions/83167/abelian-subgroups-of-ball-quotient/83172#83172 Answer by BS for Abelian subgroups of ball quotient BS 2011-12-11T10:32:35Z 2011-12-11T10:32:35Z <p>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 ${\mathbb Z}^2$. </p> <p>This is not the case, because $\Gamma$ is a Gromov-hyperbolic group <a href="http://en.wikipedia.org/wiki/Hyperbolic_group" rel="nofollow">http://en.wikipedia.org/wiki/Hyperbolic_group</a>, due to the fact that $X$ has a negatively curved riemannian metric (quotient of that of the complex hyperbolic plane, aka 4-ball).</p>