# Moscow Mathematical Journal

Volume 23, Issue 1, January–March 2023 pp. 113–120.

Homology Group Automorphisms of Riemann Surfaces

**Authors**:
Rubén A. Hidalgo (1)

**Author institution:**(1) Departamento de Matemática y Estadística, Universidad de La Frontera, Temuco, Chile

**Summary: **

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, if 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 $(0;k,\dots,k)$, 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 $(0;k_{j},\dots,k_{j})$ 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 $\mathrm{Aut}(S)$ of each homology group $A$ is also obtained.

2020 Math. Subj. Class. 30F10, 30F40.

**Keywords:**Riemann surface, automorphism, Fuchsian group.

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