Talk:Surface groups as subgroups of hyperbolic groups (Ex)
From Manifold Atlas
Let be a closed aspherical -manifold such that . Since is hyperbolic, it does not contain as a subgroup. Using Geometrisation, we can therefore conclude that is a closed hyperbolic -manifold. Then Agol's virtual fibering theorem shows that has a finite cover that is a surface bundle over . Let be the fundamental group of the fibre which is a closed orientable surface of genus . Then is a subgroup of which is a finite index subgroup of .