Mumford curves and Mumford groups in positive characteristic

Abstract

A Mumford group is a discontinuous subgroup Γ of PGL2(K), where K denotes a non archimedean valued field, such that the quotient by Γ is a curve of genus 0. As abstract group Γ 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 [26]. Part of it clarifies, corrects and extends work of G. Cornelissen, F. Kato and K. Kontogeorgis.