A note on the homology covering of closed Klein surfaces
In previous works we have seen that a finitely generated torsion-free non-elementary function group is uniquely determined by its commutator subgroup. In this note, we observe that under the presence of orientation-reversing conformal automorphisms the above rigidity property still valid. More precisely, we see that finitely generated torsion-free reversing Fuchsian groups of the first kind, without parabolic transformations, are uniquely determined by their commutator subgroup. The arguments of the proof follows the same lines as for the orientable situation.