Talk:Surface groups as subgroups of hyperbolic groups (Ex)

Let $M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_Czjn8l$$<wikitex>; Let $M$ be a closed aspherical $-manifold such that $\pi_1(M)\cong G$. Since $G$ is hyperbolic, it does not contain $\mathbb{Z}^2$ as a subgroup. Using Geometrisation, we can therefore conclude that $M$ is a closed hyperbolic $-manifold. Then Agol's virtual fibering theorem shows that $M$ has a finite cover $N$ that is a surface bundle over $S^1$. Let $\pi_1(F)$ be the fundamental group of the fibre $F$ which is a closed orientable surface of genus $g\geq 2$. Then $\pi_1(F)$ is a subgroup of $\pi_1(N)$ which is a finite index subgroup of $\pi_1(M)$. </wikitex>M$ be a closed aspherical $3$$3$-manifold such that $\pi_1(M)\cong G$$\pi_1(M)\cong G$. Since $G$$G$ is hyperbolic, it does not contain $\mathbb{Z}^2$$\mathbb{Z}^2$ as a subgroup. Using Geometrisation, we can therefore conclude that $M$$M$ is a closed hyperbolic $3$$3$-manifold. Then Agol's virtual fibering theorem shows that $M$$M$ has a finite cover $N$$N$ that is a surface bundle over $S^1$$S^1$. Let $\pi_1(F)$$\pi_1(F)$ be the fundamental group of the fibre $F$$F$ which is a closed orientable surface of genus $g\geq 2$$g\geq 2$. Then $\pi_1(F)$$\pi_1(F)$ is a subgroup of $\pi_1(N)$$\pi_1(N)$ which is a finite index subgroup of $\pi_1(M)$$\pi_1(M)$.