Inexistence de pavages mesurables invariants par un r\'eseau dans un espace homog\`ene d'un groupe de Lie simple

TL;DR

This paper proves that homogeneous spaces of almost simple Lie groups cannot be measurably tiled invariantly by a lattice, refining the Howe-Moore ergodicity theorem.

Contribution

It establishes a new non-existence result for invariant measurable tilings in homogeneous spaces of almost simple Lie groups, extending ergodic theory insights.

Findings

Homogeneous spaces of almost simple Lie groups lack invariant measurable tilings.

Refinement of the Howe-Moore ergodicity theorem.

Provides new constraints on invariant structures in Lie group actions.

Abstract

We prove that an homogeneous space of an almost simple Lie group does not have any measurable tiling invariant by a lattice of the Lie group. This refines the Howe-Moore ergodicity theorem. -- On d\'emontre qu'un espace homog\`ene d'un groupe de Lie presque simple n'admet pas de pavage mesurable invariant par un r\'eseau du groupe de Lie. Ceci constitue un raffinement du th\'eor\`eme d'ergodicit\'e de Howe-Moore.