Rigidity of surface groups I am interested in the following result:

Theorem: A torsion-free group which contains the fundamental group of a closed surface as a finite index must be the fundamental group of a closed surface. 

This is a consequence of a harder theorem stating that PD²-groups are surface groups; see Eckmann's survey, Poincaré duality groups of dimension two are surface groups. Similar kind of rigidity appears for free groups and abelian free groups. 
Do you know if alternative proofs exist? 

You can also use Tukia's theorem, from the 1980's, which shows that a uniform convergence subgroup of the homeomorphism group of the circle is either conjugate to a Fuchsian group or is one of an explicitly described class of groups which are not torsion free. Your group does indeed have a uniform convergence action on the circle, and the kernel must be finite, and so the theorem does apply to your group.
See Tukia's paper "Homeomorphic conjugates of Fuchsian groups" in J. Reine Angew. Math.
There's a history behind these kinds of theorems which goes back to Nielsen, but early work had some errors. Zieschang was the first to work out some cases with full rigor (and the one who identifed the errors in early works), and I believe that this paper of Tukia may be the first to rigorously cover your case of interest.
