Joinings classification and applications [after Einsiedler and Lindenstruass]

TL;DR

This paper surveys the classification of joinings for higher-rank torus actions on arithmetic quotients, highlighting their applications in solving arithmetic problems and introducing a new application for higher rank simple groups.

Contribution

It provides a comprehensive overview of Einsiedler and Lindenstrauss's work on joinings classification and demonstrates its applications in arithmetic and new group contexts.

Findings

Classification of joinings for higher-rank torus actions

Applications to arithmetic problems

Introduction of a new application for higher rank simple groups

Abstract

This is the text accompanying my Bourbaki seminar on the work of Einsiedler and Lindenstrauss on joinings. The first five sections surveys their proof of the classification of joinings of higher-rank torus actions on arithmetic quotients of semisimple or perfect algebraic groups. The last section surveys how this classification can be used to tackle arithmetic problems, together with a new application for higher rank simple groups.