Arithmeticity of discrete subgroups containing horospherical lattices

Abstract

Let $G$ be a semisimple real algebraic Lie group of real rank at least $2$, and let $U$ be the unipotent radical of a nontrivial 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 works of Selberg and Oh.