Schottky Groups over Valuation Rings

TL;DR

This paper extends the theory of Schottky groups to valuation rings over complete valued fields, constructing associated $b1$-tree spaces and analyzing their properties, including group actions and quotient graphs.

Contribution

It introduces a new framework for defining hyperbolic matrices and Schottky groups over valuation rings, generalizing classical non-archimedean cases.

Findings

Constructed $b1$-tree spaces analogous to Bruhat-Tits trees.

Defined hyperbolic matrices and Schottky groups over valuation fields.

Established a method to build such groups and analyze their quotient graphs.

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.