Schottky Groups over Valuation Rings

Abstract

Given a non-trivial complete valued field $K$ with value group $\Lambda$, we construct a $\Lambda$-tree space associated to $K$ analog of the Bruhat-Tits tree, and locally finite trees associated to compact subsets of the projective line. We propose a definition of hyperbolic matrix and Schottky group over such field $K$. To any such Schottky group $\Gamma$, we associate a compact set with an action of $\Gamma$, such that the quotient graph of the associated tree is a finite graph, and $\Gamma$ is identified with its fundamental group. Finally explain a method to construct such groups. This results extend the classical ones for discrete valuations of Mumford and non-archimedean rank 1 valuations of Gerritzen and Van der Put.