Branch points of split degenerate superelliptic curves I: construction of Schottky groups

Let $K$ be a field with a discrete valuation, and let $p$ be a prime. It is known that if $\mathrm{\Gamma }⊲{\mathrm{\Gamma }}_0<{\mathrm{P}\mathrm{G}\mathrm{L}}_2(K)$ 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 $\mathrm{\Gamma }$ can be algebraized as a superelliptic curve $C:y^p=f(x)\mathrm{/}K$. The subset $S\subset K\cup \{\mathrm{\infty }\}$ consisting of these pairs $a_i,b_i$ of fixed points is mapped modulo $\mathrm{\Gamma }$ to the set of branch points of the superelliptic map $x:C\to {\mathbb{P}}_K^1$. We produce an algorithm for determining whether an input even-cardinality subset $S\subset K\cup \{\mathrm{\infty }\}$ consists of fixed points of generators of such a group ${\mathrm{\Gamma }}_0$ and which, in the case of a positive answer, modifies $S$ into a subset $S^{\mathrm{m}\mathrm{i}\mathrm{n}}\subset K\cup \{\mathrm{\infty }\}$ with particularly nice properties. Our results do not involve any restrictions on the prime $p$ or on the residue characteristic of $K$ and allow these to be the same.