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

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Let k be a global field and let kᵥ 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 kᵥ-rank 1. Let G=G(kᵥ). Let Γbe 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 by determining the structure of C. It is shown that C is a free profinite product, one of whose factors is Fω, 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=SL₂(O(S)), where O(S) is 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 Γon the Bruhat-Tits tree associated with G.