A joining classification and a special case of Raghunathan's conjecture in positive characteristic (with an appendix by Kevin Wortman)

TL;DR

This paper classifies joinings for maximal horospherical subgroups on homogeneous spaces in positive characteristic and deduces a special case of Raghunathan's orbit closure conjecture, also characterizing quasi-isometries of higher rank lattices.

Contribution

It provides the first unrestricted classification of joinings in positive characteristic and links it to a special case of Raghunathan's conjecture, with an appendix on quasi-isometries.

Findings

Classification of joinings without characteristic restrictions

A special case of Raghunathan's orbit closure conjecture proved

Quasi-isometries of higher rank lattices characterized

Abstract

We prove the classification of joinings for maximal horospherical subgroups acting on homogeneous spaces without any restriction on the characteristic. Using the linearization technique we deduce a special case of Raghunathan's orbit closure conjecture. In the appendix quasi-isometries of higher rank lattices in semisimple algebraic groups over fields of positive characteristic are characterized.