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-Fermat curves: an embedding of moduli spaces
A group , where are integers, of
conformal automorphisms of a closed Riemann surface is called a
-Fermat group if it acts freely with quotient of genus . We
study some properties of these type of objects, in particular, we observe that
is non-hyperelliptic and, if , where is a prime
integer and , then is the unique -Fermat group of . Let
be a co-compact torsion free Fuchsian group such that . If is its normal subgroup generated by its
commutators and the -powers of its elements, then there is a biholomorphism
between and congugating to
. The inclusion induces a natural
holomorphic embedding of the corresponding Teichm\"uller spaces. Such an
embedding induces a holomorphic map, at the level of their moduli spaces,
. As a consequence
of the results on -Fermat groups, we provide sufficient conditions for
the injectivity of .Comment: This is an actualized expanded version. Some changes of sections have
been don
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