$(g,k)$-Fermat curves: an embedding of moduli spaces

A group $H \cong {\mathbb Z}_{k}^{2g}$, where $g,k \geq 2$ are integers, of conformal automorphisms of a closed Riemann surface $S$ is called a $(g,k)$-Fermat group if it acts freely with quotient $S/H$ of genus $g$. We study some properties of these type of objects, in particular, we observe that $S$ is non-hyperelliptic and, if $k=p^{r}$, where $p>84(g-1)$ is a prime integer and $r \geq 1$, then $H$ is the unique $(g,k)$-Fermat group of $S$. Let $\Gamma$ be a co-compact torsion free Fuchsian group such that $S/H={\mathbb H}^{2}/\Gamma$. If $\Gamma_{k}$ is its normal subgroup generated by its commutators and the $k$-powers of its elements, then there is a biholomorphism between $S$ and ${\mathbb H}^{2}/\Gamma_{k}$ congugating $H$ to $\Gamma/\Gamma_{k}$. The inclusion $\Gamma_{k} < \Gamma$ induces a natural holomorphic embedding $\Theta_{k}:{\mathcal T}(\Gamma) \hookrightarrow {\mathcal T}(\Gamma_{k})$ of the corresponding Teichm\"uller spaces. Such an embedding induces a holomorphic map, at the level of their moduli spaces, $\Phi_{k}:{\mathcal M}(\Gamma) \to {\mathcal M}(\Gamma_{k})$. As a consequence of the results on $(g,k)$-Fermat groups, we provide sufficient conditions for the injectivity of $\Phi_{k}$.