Homology group automorphisms of Riemann surfaces

Abstract

If $\Gamma$ is a finitely generated Fuchsian group such that its derived subgroup $\Gamma'$ is co-compact and torsion free, then $S={\mathbb H}^{2}/\Gamma'$ is a closed Riemann surface of genus $g \geq 2$ admitting the abelian group $A=\Gamma/\Gamma'$ 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, is there are different Fuchsian groups $\Gamma_{1}$ and $\Gamma_{2}$ with $\Gamma_{1}'=\Gamma'_{2}$? It is known that if $\Gamma_{1}$ and $\Gamma_{2}$ are both of the same signature $(0;k,\ldots,k)$, for some $k \geq 2$, then the equality $\Gamma_{1}'=\Gamma_{2}'$ ensures that $\Gamma_{1}=\Gamma_{2}$. Generalizing this, we observe that if $\Gamma_{j}$ has signature $(0;k_{j},\ldots,k_{j})$ and $\Gamma_{1}'=\Gamma'_{2}$, then $\Gamma_{1}=\Gamma_{2}$. We also provide examples of surfaces $S$ with different homology groups. A description of the normalizer in ${\rm Aut}(S)$ of each homology group $A$ is also obtained.