Schottky groups in valuation rings

Abstract

Given a complete field $K$, and a valuation over $K$, we can construct a "tree of balls", where the vertex are the open balls obtained from a subset $\mathcal{L}$ of $\mathbb{P}^{1}(K)$ seen as a topological space, and the edges are obtained from the paths between elements of the subset of $\mathbb{P}^{1}(K)$. In order to define the open balls we need to give a topology. It comes from our valuation and gives the property that or two balls does not intersect or one is contained in the other. Moreover given a Schottky group $\Gamma$ acting on the tree of balls we will see that we obtain a finite tree. In order to see that we will see first that this tree of balls of a subset $\mathcal{L}$ of $\mathbb{P}^{1}(K)$ is locally finite. We will see that the subset $\mathcal{L}$ has to be compact in order to guarantee the finiteness of the resultant tree. Other result will consist on see that the closure of the limit points of a Schottky group, $\mathcal{L}_{\Gamma}$, is equal to the closure of the orbit of some point, which by definition of Schottky group will guarantee that this set is compact so we will be able to apply the previous theory. In order to define a Schottky group we will consider that it has to be topologically nilpotent in order to extend the non-Archimedian results to any totally ordered group as a image of our valuation. We also will see a characterisation of hyperbolic matrices and we will consider some example of the graph $\mathcal{L}_{\Gamma}/\Gamma$.