The congruence subgroup problem

Abstract

Abstract. This is a short survey of the progress on the congruence subgroup problem since the sixties when the first major results on the integral unimodular groups appeared. It is aimed at the non-specialists and avoids technical details.

Bak's work on theK-theory of rings

“…Here we take a very limited view and do not attempt to discuss the subsequent remarkable results by many authors. To get a broader prospective, the reader should consult the book by Platonov and Rapinchuk [117] and the recent papers by Prasad, Raghunathan [122] and Rapinchuk.…”

Arithmetic groups have rational representation growth

“…It is known that "most" irreducible lattices in Lie groups of rank ≥ 2 have the congruence subgroup property, and a conjecture of Serre asserts that all of them do. See [20] for a survey on the congruence subgroup property.…”

Arithmeticity of discrete subgroups containing horospherical lattices

“…Moreover, by a previous result of Raghunathan and Venkataramana in [28] and [33], it was known that Conjecture 1.1 was true for a subgroup of an arithmetic group. Indeed, their results say that if G is an absolutely Q-simple algebraic group of real rank at least two, and U, U − is a pair of non-trivial opposite maximal horospherical subgroups defined over Q then any subgroup Γ of G Z that contains lattices in both U and U − has finite index in G Z .…”

“…This proposition is proven in [28] and [33] There exists a finite index subgroup ∆ L ⊂ L Z that normalizes both ∆ and ∆ − . By [28] and [33], the group generated by the lattices (∆ L ∩ V )∆ and (∆ L ∩ V − )∆ − of V Z and V − Z has finite index in G Z and normalizes Γ. Therefore, by the Margulis normal subgroup theorem, Γ has finite index in G Z .…”

“…The following result of Raghunathan and Venkataramana in [28] and [33] says that in order to prove Theorem 2.16 for U reflexive, we only have to find a Q-structure on G such that Γ ⊂ G Q . Proof.…”