Proper actions on corank-one reductive homogeneous spaces

TL;DR

This paper characterizes all torsion-free discrete subgroups acting properly discontinuously on corank-one reductive homogeneous spaces over local fields, providing a general framework for understanding such actions in algebraic and geometric contexts.

Contribution

It introduces a comprehensive description of torsion-free discrete groups acting properly on corank-one reductive homogeneous spaces, extending previous understanding in the setting of local fields.

Findings

Classification of torsion-free discrete subgroups acting properly

A general result on Cartan projections for these groups

Application to corank-one reductive homogeneous spaces

Abstract

Let k be a local field and G the set of k-points of a connected semisimple algebraic k-group of rank one. We describe all torsion-free discrete subgroups of G\times G acting properly discontinuously on G by left and right multiplication. To this end, we prove a general result on the Cartan projection of discrete groups acting properly discontinuously on corank-one reductive homogeneous spaces over k.