On systems of generators of arithmetic subgroups of higher rank groups

Abstract

We show that any two maximal disjoint unipotent subgroups of an irreducible non-cocompact lattice in a Lie group of rank atleast two generates a lattice. The proof uses techniques of the solution of the congruence subgroup problem. We show that any two maximal opposing unipotent subgroups of an irreducible lattice in a higher rank Lie Group, generate a lattice in the Lie Group. The method of proof is to use certain techniques of the solution of the congruence subgroup problem of arithmetic lattices in higher rank groups. We freely use the notation and results of [3] without giving explicit references therein. Let G be a simply connected absolutely almost simple linear algebraic group defined and isotropic over a global field K. Let C/+ be the unimpotent radical (which is defined over K) of a minimal parabolic A'-subgroup P+ of G. Let U~~ be the unipotent radical of another minimal parabolic A'-subgroup P~ of G which is opposed to P+ in the sense that U+ Γ\U~ = {1}. Let S be a finite set of places of including all the archimedian ones, if any. We call thering A — Os — {x G K \x v < 1 for all places v of A', not in S} the ring of S-integers in K. Choose a faithful representation G <—» GLpj defined over and define G(Os) = {g G G gij G Os, 1 < i,j < N}. The subgroups in G which are of finite index in G(Os) are called S-arithmetic groups. Define the 5-rank of G to be the sum Kv — rank(G). Given a non-zero ideal α of A and an algebraic ves