Hyperbolic Surface Subgroups of Doubles

Let us consider the presentation complex X of an arbitrary 2-generator presentation. We puncture each 2-cell and double along the holes to get Y, called the double of X. We show that the fundamental group D of Y contains a closed hyperbolic surface group or D splits as a free product (and hence reduces to a simpler case). With more generators, we also resolve the case when the relator set is minimal and k-regular.