HOMOLOGY GROUP AUTOMORPHISMS OF RIEMANN SURFACES

If & UGamma; is a finitely generated Fuchsian group such that its derived subgroup & UGamma;' is co-compact and torsion free, then S =H2/& UGamma;' is a closed Riemann surface of genus g ?, 2 admitting the abelian group A = & UGamma;/& UGamma;' as a group of conformal automorphisms. We say that A is a homology group of S. A natural question is if S admits unique homology groups or not, in other words, if there are different Fuchsian groups & UGamma;1 and & UGamma;2 with & UGamma;'1 = & UGamma;'2? It is known that if & UGamma;1 and & UGamma;2 are both of the same signature (0; k, ... , k), for some k ?, 2, then the equality & UGamma;'1 = & UGamma;'2 ensures that & UGamma;1 = & UGamma;2. Generalizing this, we observe that if & UGamma;j has signature (0; kj, ..., kj) and & UGamma;'1 = & UGamma;'2, then & UGamma;1 = & UGamma;2. We also provide examples of surfaces S with different homology groups. A description of the normalizer in Aut(S) of each homology group A is also obtained.