Homology coverings of Riemann surfaces (Q1319191)

We extend a result on homology coverings of closed Riemann surfaces due to Maskit to the class of analytically finite ones. We show that if \(S\) is an analytically finite hyperbolic Riemann surface, then its conformal structure is determined by the conformal structure of its homology cover. The homology cover of a Riemann surface \(S\) is the highest regular covering of \(S\) with an Abelian group of covering transformations. In fact, we show that the commutator subgroup of any torsion-free, finitely generated Fuchsian group of the first kind determines it uniquely.