Arithmeticity of discrete subgroups containing horospherical lattices

TL;DR

This paper proves that certain discrete subgroups in higher-rank Lie groups containing horospherical lattices are necessarily arithmetic, confirming a conjecture of Margulis and extending prior results.

Contribution

It establishes the arithmeticity of Zariski dense subgroups containing horospherical lattices in higher-rank semisimple Lie groups, solving a longstanding conjecture.

Findings

Discrete subgroups with horospherical lattices are arithmetic.

Confirms Margulis's conjecture on arithmeticity.

Extends previous work by Hee Oh.

Abstract

Let $G$ be a semisimple real algebraic Lie group of real rank at least two and $U$ be the unipotent radical of a non-trivial parabolic subgroup. We prove that a discrete Zariski dense subgroup of $G$ that contains an irreducible lattice of $U$ is an arithmetic lattice of $G$. This solves a conjecture of Margulis and extends previous work of Hee Oh.