  Consider the moduli space $\mathcal{M}_{g}$ of Riemann surfaces of genus
$g\geq 2$ and its Deligne-Munford compactification $\bar{\mathcal{M}_{g}}$. We
are interested in the branch locus ${\mathcal{B}_{g}}$ for $g>2$, i.e., the
subset of $\mathcal{M}_{g}$ consisting of surfaces with automorphisms. It is
well-known that the set of hyperelliptic surfaces (the hyperelliptic locus) is
connected in $\mathcal{M}_{g}$ but the set of (cyclic) trigonal surfaces is
not. By contrast, we show that for $g\geq 5$ the set of (cyclic) trigonal
surfaces is connected in $\bar{\mathcal{M}_{g}}$. To do so we exhibit an
explicit nodal surface that lies in the completion of every equisymmetric set
of 3-gonal Riemann surfaces. For $p>3$ the connectivity of the $p$-gonal loci
becomes more involved. We show that for $p\geq 11$ prime and genus $g=p-1$
there are one-dimensional strata of cyclic $p$-gonal surfaces that are
completely isolated in the completion $\bar{\mathcal{B}_{g}}$ of the branch
locus in $\bar{\mathcal{M}_{g}}$.
