Mumford curves and Mumford groups in positive characteristic

TL;DR

This paper classifies Mumford groups in positive characteristic, constructs Mumford curves with large automorphism groups, and introduces orbifolds in rigid spaces, extending previous work on discontinuous subgroups of PGL(2,K).

Contribution

It provides a classification of certain Mumford groups in positive characteristic and establishes bounds on automorphism groups of Mumford curves, extending the theory of discontinuous subgroups.

Findings

Classified amalgams with two or three branch points in positive characteristic.

Discovered Mumford curves with large automorphism groups.

Established upper bounds for automorphism group orders using combinatorial methods.

Abstract

A Mumford group is a discontinuous subgroup $\Gamma$ of PGL(2,K), where K denotes a non archimedean valued field, such that the quotient by $\Gamma$ is a curve of genus 0. As abstract group $\Gamma$ is an amalgam of a finite tree of finite groups. For K of positive characteristic the large collection of amalgams having two or three branch points is classified. Using these data Mumford curves with a large group of automorphisms are discovered. A long combinatorial proof, involving the classification of the finite simple groups, is needed for establishing an upper bound for the order of the group of automorphisms of a Mumford curve. Orbifolds in the category of rigid spaces are introduced. For the projective line the relations with Mumford groups and singular stratified bundles are studied. This paper is a sequel to our paper "Discontinuous subgroups of PGL(2,K)" published in Journ. of Alg. (2004). Part of it clarifies, corrects and extends work of G.~Cornelissen, F.~Kato and K.~Kontogeorgis.