  The moduli space $\mathcal{M}_{g}$ of compact Riemann surfaces of genus $g$
has orbifold structure, and the set of singular points of such orbifold is the
\textit{branch locus} $\mathcal{B}_{g}$. Given a prime number $p \ge 7$,
$\mathcal{B}_{g}$ contains isolated strata consisting of $p$-gonal Riemann
surfaces for genera $g \ge {3(p-1)\over 2}$,that are multiple of ${p-1 \over
2}$. This is a generalization of the results obtained in \cite{[BCI1]} for
pentagonal Riemann surfaces, and the results of \cite{[K]} and \cite{[CI3]} for
zero- and one-dimensional isolated strata in the branch locus.
