The aim of this course will be to prove that any Zariski dense discrete subgroup of a higher rank simple Lie group which intersects cocompactly the unipotent radical of a proper parabolic subgroup is an arithmetic lattice. This work with Miquel extends previous work of Selberg and Hee Oh and answers an old question of Margulis.

The course will focus on concrete cases like SL(n,R) or SO(p,q) and will be a journey in the world of discrete subgroups of Lie groups. We will explain how classical tools and new techniques enter the proof: Zassenhauss neighborhoods, Auslander theorem, Margulis lemma, Bruhat decomposition, Borel-Harish-Chandra finiteness theorem, Borel density theorem, Raghunathan-Venkataramana congruence subgroup theorem, Weil rigidity theorem, Ratner topological theorem etc.