  The fundamental group of Fermat and generalized Fermat curves is computed.
These curves are Galois ramified covers of the projective line with abelian
Galois groups $H$. We provide a unified study of the action of both cover
Galois group $H$ and the absolute Galois group $\mathrm{Gal}(\bar{\Q}/\Q)$ on
the pro-$\ell$ homology of the curves in study. Also the relation to the
pro-$\ell$ Burau representation is investigated.
