Talk:Fundamental groups of surfaces (Ex)
From Manifold Atlas
Let be a surface. Suppose . Then is finite and hence hyperbolic.
Suppose . Then the interior of is a quotient of by a discrete torsion-free subgroup of . Then is hyperbolic since it is isomorphic to which acts freely and properly discontinuously on .
Now let . If has boundary, then it is homeomorphic to an annulus or a Mobius band. Each of which have fundamental group which is hyperbolic since its Cayley graph is quasi-isometric to a tree.
The only remaining cases are the Klein bottle and the torus. The latter has fundamental group , while the fundamental group of the former contains as an index 2 subgroup. Thus neither of these groups is hyperbolic.