Let $G$ be a co-compact torsion free Fuchsian group of genus $g \geq2$ and,
for each integer $k \geq 2$, $G_{k}$ be its normal subgroup generated by the
$k$-powers of the elements of $G$ together its commutators. There is a natural
holomorphic embedding $\Theta_{k}:{\mathcal T}(G) \hookrightarrow {\mathcal
T}(G_{k})$ of the corresponding Teichm\"uller spaces. If $\pi:{\mathcal T}(G)
\to {\mathcal M}(G)$ and $\pi_{k}:{\mathcal T}(G_{k}) \to {\mathcal M}(G_{k})$
are the corresponding (branched) Galois covers over their moduli spaces, then
there is a holomorphic map $\Phi_{k}:{\mathcal M}(G) \to {\mathcal M}(G_{k})$
such that $\pi_{k} \circ \Theta_{k}=\Phi_{k} \circ \pi$. The aim of this paper
is to investigate where this last map is injective or not.