Surface Subgroups of Kleinian Groups with Torsion

A subgroup of a group is known as a surface subgroup if it is isomorphic to the fundamental group of a closed connected orientable surface with positive genus. The surface subgroup conjecture in low-dimensional topology proposes that if the fundamental group of a closed orientable irreducible 3-manifold is infinite, then it contains a surface subgroup. With the recent solution to the geometrisation conjecture by Perelman ([19], [20], [21]), it suffices to prove the surface subgroup conjecture for closed orientable hyperbolic 3-manifolds. Gromov has asked a much more general question [1]: does every infinite, word hyperbolic group that is not virtually free contain a surface subgroup? In this paper, we will establish this conjecture for a large class of Kleinian groups. By a Kleinian group, we mean a discrete subgroup of PSL(2,C).