Quotients compacts des groupes ultrametriques de rang un

TL;DR

This paper characterizes all finitely generated torsion-free discrete subgroups of a product of rank-one semisimple algebraic groups over nonarchimedean fields, and shows their stability under small deformations in terms of proper discontinuity and cocompactness.

Contribution

It provides a complete description of certain discrete subgroups acting on ultrametric symmetric spaces and demonstrates their stability under small deformations.

Findings

Classification of all such discrete subgroups

Stability of group actions under small deformations

Proper discontinuity and cocompactness are preserved

Abstract

Let G be the set of k-points of a connected semisimple algebraic group of k-rank one over a nonarchimedean local field k. We describe all finitely generated torsion-free discrete subgroups of G\times G acting properly discontinuously and cocompactly on G by left and right multiplication. We prove that after a small deformation in G\times G such a discrete subgroup keeps acting freely, properly discontinuously, and cocompactly on G. ----- Soit G l'ensemble des k-points d'un groupe algebrique semi-simple connexe de k-rang un sur un corps local ultrametrique k. Nous decrivons tous les sous-groupes discrets de type fini sans torsion de G\times G qui agissent proprement et cocompactement sur G par multiplication a gauche et a droite. Nous montrons qu'apres une petite deformation dans G\times G un tel sous-groupe discret agit encore librement, proprement et cocompactement sur G.