Arithmeticity of discrete subgroups containing horospherical lattices

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.

Arithmeticity of discrete subgroups

“…This theorem is the main result of my article [3] with Sébastien Miquel. It solves a conjecture of Margulis.…”

“…Here is the general version of Proposition 3.8 that applies to all of the groups in Theorem 1.1 and whose proof can be found in [3,Proposition 4.11]. PROPOSITION 3.9.…”

“…13 is classical when is a lattice in G. Note that in Theorem 1.1, we use this definition for a discrete subgroup of G which is not a lattice in G but a lattice in U (see [3,Lemma 4.3] for more insight on this notion). Note also that this irreducibility condition is always satisfied when G is quasisimple.…”

“…For more insight on the case where G is not simple, the reader should consult [3, §4.6, 5, 16] when G is a product of rank-one Lie groups. In general, one has to follow reduction steps, see [3,Ch. 5], where one keeps track of the assumption that is irreducible.…”

“…Most of the students in the audience were either graduate students or postdocs. I tried to keep the informal style of the lectures, giving only complete proof of representative examples, focusing on the main ideas, pointing out those ideas that are often useful in this subject, recalling shortly the proof of preliminary classical results and leaving the technical issues to my joint paper [3] with Sébastien Miquel.…”