# Schottky groups, Mumford curves and $p$-adic uniformization

Let $K$ be a $p$-adic field and $\Gamma$ be Schottky group of $g$ generators and $L \subset \mathbb{P}^1_K$ be the limit set of $\Gamma$. Let $\mathbb{P}^1_K - L= \Omega$ and we know that there exists a covering of $\Omega$ such that the reduction of $\Omega$ is a tree of projective lines whose dual graph is isomorphic to the tree $T(\Gamma)$.

Assume that quotient $T(\Gamma)/\Gamma$ is given by two vertices and $g+1$ edges. Let $v$ be a vertex of $T(\Gamma)$ and $r : \Omega \rightarrow T(\Gamma)$, can we describe the open $r^{-1}(v)$?

$T(\Gamma)$ is constructed by gluing semi-stable skeletons and hence the inverse image of any vertice $v$ is the closed unit disc punctured by finitely many maximal open rational discs and the number of these maximal open discs corresponds to the index of $v$ and it is equal to $g+1$ since $T(\Gamma)$ is the universel covering of $T(\Gamma)/\Gamma$ and $T(\Gamma)/\Gamma$ has $g+1$ edges.