Finitely generated profinitely dense free groups in higher rank semi-simple groups

Abstract

We prove that if Γ is an arithmetic subgroup of a non-compact linear semi-simple group G such that the associated simply connected algebraic group over Q has the so-called congruence subgroup property, then Γ contains a finitely generated profinitely dense free subgroup. As a corollary we obtain a f·g·p·d·f subgroup of SL_n (Z) (n ≧ 3). More generally, we prove that if Γ is an irreducible arithmetic non-cocompact lattice in a higher rank group, then Γ contains f·g·p·d·f groups.