Discrete subgroups generated by lattices in opposite horospherical subgroups of $SL(3,\mathbb{C})$ following Hee Oh

TL;DR

This paper proves that a discrete subgroup generated by two lattices in opposite minimal horospherical subgroups of SL(3,C) is arithmetic, extending previous results from real to complex groups using orbit and rational structure analysis.

Contribution

It extends the known results on lattices in horospherical subgroups from real to complex groups, specifically for SL(3,C), using orbit closure and rational form techniques.

Findings

Generated subgroup is arithmetic

Orbit closure characterized by Ratner's theorem

Method extends to complex Lie groups

Abstract

We prove that a discrete subgroup generated by two lattices in opposite minimal horospherical subgroups of $SL(3,\mathbb{C})$ is arithmetic and thus by a Borel and Harish-Chandra also a lattice. We follow the method and ideas used by Oh in \cite{Ref13}. There, Oh proves the same result for lattices in minimal horospherical subgroups of $SL(n,\mathbb{R})$. The method consists of studying the orbits of the generating lattices in the horospherical subgroups under the conjugation action of the commutator of the corresponding Levi subgroup in order to obtain a rational structure for $SL(3,\mathbb{C})$. A theorem of Ratner reduces the possibilities for the closure of the orbits. Then, by the discreteness of the generated subgroup, it can be shown the closeness of the orbits. Finally, using this information, it's possible to find a rational form such that the subgroup generated by the lattices is arithmetic.