  Let $K$ be a field with a discrete valuation, and let $p$ be a prime. It is
known that if $\Gamma \lhd \Gamma_0 < \mathrm{PGL}_2(K)$ is a Schottky group
normally contained in a larger group which is generated by order-$p$ elements
each fixing $2$ points $a_i, b_i \in \mathbb{P}_K^1$, then the quotient of a
certain subset of the projective line $\mathbb{P}_K^1$ by the action of
$\Gamma$ can be algebraized as a superelliptic curve $C : y^p = f(x) / K$. The
subset $S \subset K \cup \{\infty\}$ consisting of these pairs $a_i, b_i$ of
fixed points is mapped bijectively modulo $\Gamma$ to the set $\mathcal{B}$ of
branch points of the superelliptic map $x : C \to \mathbb{P}_K^1$. A conjecture
of Gerritzen and van der Put, in the case that $C$ is hyperelliptic and $K$ has
residue characteristic $\neq 2$, compares the cluster data of $S$ with that of
$\mathcal{B}$. We show that this conjecture requires a slight modification in
order to hold and then prove a much stronger version of the modified conjecture
that holds for any $p$ and any residue characteristic.
