A non-homogeneous orbit of a diagonal subgroup

TL;DR

This paper constructs specific examples of lattice orbits in SL(n,R) that challenge a conjecture by Margulis, showing non-homogeneous orbit closures that do not factor through one-parameter non-unipotent group actions.

Contribution

It provides explicit counterexamples to a conjecture of Margulis by constructing non-homogeneous orbit closures in SL(n,R) for n>5.

Findings

Counterexamples to Margulis's conjecture.

Orbit closures that are non-homogeneous.

Orbit closures not factoring through one-parameter non-unipotent groups.

Abstract

Let G=SL(n,R) with n>5. We construct examples of lattices Gamma of G, subgroup A of the diagonal group and points x in G/Gamma such that the closure of the orbit Ax is not homogeneous but does not factors through the action of a one-parameter non-unipotent group. This contradicts a conjecture of Margulis.