Homology group automorphisms of Riemann surfaces

TL;DR

This paper investigates the uniqueness of homology groups arising from Fuchsian groups acting on Riemann surfaces and explores conditions under which different groups produce the same surface automorphisms.

Contribution

It generalizes known results on homology group uniqueness for specific signatures and provides examples of surfaces with different homology groups, along with their normalizers.

Findings

Uniqueness of homology groups for certain signatures

Existence of surfaces with multiple homology groups

Description of normalizers of homology groups in automorphism groups

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.