The congruence kernel of an arithmetic lattice in a rank one algebraic group over a local field

Abstract

Let k be a global field and let k v be the completion of k with respect to v, a non-archimedean place of k. Let G be a connected, simply-connected algebraic group over k, which is absolutely almost simple of k v -rank 1. Let G = G(k v ). Let Γ be an arithmetic lattice in G and let C = C(Γ) be its congruence kernel. Lubotzky has shown that C is infinite, confirming an earlier conjecture of Serre. Here we provide complete solution of the congruence subgroup problem for Γ by determining the structure of C. It is shown that C is a free profinite product, one of whose factors isF ω , the free profinite group on countably many generators. The most surprising conclusion from our results is that the structure of C depends only on the characteristic of k. The structure of C is already known for a number of special cases. Perhaps the most important of these is the (non-uniform) example Γ = SL 2 (O(S)), where O(S) is the the ring of S-integers in k, with S = {v}, which plays a central role in the theory of Drinfeld modules. The proof makes use of a decomposition theorem of Lubotzky, arising from the action of Γ on the Bruhat-Tits tree associated with G.

Subgroups of algebraic groups which are clopen in the S-congruence topology

“…One important consequence of Lemma 2.1 is that Lemma 5.7 in [8] is true for all q so that, in the terminology of [8], the principal result always holds. This leads to a significant simplification in the proofs of [8].…”

“…This leads to a significant simplification in the proofs of [8]. Specifically Zel'manov's solution [14] of the restricted Burnside problem for topological groups is no longer required.…”

“…More precisely it is known [8,Lemma 3.7] that in this situation every S-arithmetic lattice contains lattices of the same type with arbitrarily large first Betti numbers.…”

“…What first attracted our attention to this result is that it provides a significant simplification in the proof of one of the principal results in a recent paper [8]. A result involving a subgroup which is clopen with respect to the S-congruence topology is central to Weisfeiler's celebrated work [13] on the strong approximation theorem.…”

On the Congruence Kernel for Simple Algebraic Groups

“…It is known (cf. Proposition 2.9) that in this case the group C is not finitely generated; for its precise structure in certain cases see [21], [27], [28], [71], [72]. Lubotzky [24] showed however that the congruence kernel C is always finitely generated as a normal subgroup of p Γ.…”