The Margulis-Zimmer Conjecture Definition. Let K be a global field, O its ring of integers, and let G be an absolutely simple, simply connected algebraic group defined over K. Let V be the set of all inequivalent valuations on K, and let V V denote the archimedean ones. For a subset V S V, let O S K be, as usual, the ring of S-integers in K, and let Γ < G(K) be an S-arithmetic group, namely, a subgroup commensurable with G(O S ). Then any S -arithmetic subgroup Λ < Γ (V S S) is commensurated by Γ, and we call such Λ a standard commensurated subgroup. Say that the S-arithmetic group Γ has a standard description of commensurated subgroups if every Λ < Γ commensurated by it, is standard or finite.
The Commensurated Subgroup Problem -CmSP: Let K, G, S, Γ as before, and assume that S-rank(Γ) := Σ ν S K ν -rank(g(k ν )) 2. Does Γ have a standard description of commen subgroups? The Margulis-Zimmer Conjecture: Yes. Venkataramana ( 87): Yes for Γ = G(Z) when G is defined over Q (not necessarily absolutely simple) with Q-rank 2. Inherent difficulty: There is no general way to pass from a group to a finite index subgroup in such a result: A priori it needs to be proved again for each finite index subgroup. In Venkataramana s case this calls for a use of the CSP (trivial congruence kernel?) In a joint work with George Willis we establish a stronger (fixed point) result, which is stable under commensurability, in the case of arithmetic subgroups of Chevalley groups G. It enables reducing the CmSP to Margulis NST, as well as to prove the CmSP for all the S-arithmetic groups containing the given arithmetic group.
Definition. 1. Say that a group Γ has the inner commensuratornormalizer property, if every commensurated subgroup Λ < Γ is almost normal in the sense that such Λ is commensurable in Γ with a normal subgroup of Γ. 2. Say that Γ has the outer commensurator-normalizer property if the following holds: for any group and any homomorphism ϕ : Γ, any subgroup Λ < which is commensurated by (the conjugation action of) ϕ(γ), is almost normalized by ϕ(γ), namely, a subgroup commensurable with Λ in is normalized by ϕ(γ). Theorem. Retain the notations in the CmSP and assume G is a Chevalley (i.e. split) group over K (char(k) = 0). In case G = SL 2 assume further that K is not Q or an imaginary quadratic extension of it. Then: 1. Any group commensurable with G(O) has the outer commensu normalizer property. 2. For any V S V, any S-arithmetic subgroup of G(K) has standard description of commensurated subgroups.
Theorem. Retain the assumptions and notations of previous Theorem. Assume that K admits a field embedding into R. Let A f denote the ring of finite adeles of K, G(A f ) denote, as usual, the restricted direct product of G(K ν ) over the finite ν V, and consider an S-arithmetic subgroup Γ < G(K) identified with its image in G(A f ) through the diagonal embedding. Let ϕ : Γ H be any homomorphism into an arbitrary locally compact totally disconnected group H. Then one, and exactly one of the following occurs: either 1. Im ϕ is discrete and Ker ϕ is finite (central), in which case ϕ doesn t extend, or 2. The homomorphism ϕ extends to a continuous homomorphism of the closure Γ < G(A f ) onto the closure ϕ(γ) < H. Furthermore, if the normalizer of any compact open subgroup of H is compact (or even merely amenable), then 2 necessarily holds. This Theorem uses and implies the CSP, CmSP, NST!
Deligne s construction. Let K be any number field, A its ring of adeles, and let G be any absolutely simple, simply connected algebraic group defined over K. In 96 (IHES Publ.) Deligne constructs a central extension G(A) of G(A) by the finite group µ of roots of unity in K, which splits over G(K): G(K) s d < p 1 >µ >G(A) > G(A) > 1 > Here d is the standard diagonal embedding of G(K) in G(A), and s is the splitting, that is, p s = d. Prop. Let Γ < G(K) be an infinite commensurated (e.g. S-arithmetic) subgroup. Then there exists a minimal (in the strong sense) open subgroup of G(A) which contains the subgroup s(γ). It will be denoted G Γ.
Conjecture. Assume that Γ < G(K) is a commensurated (e.g. S-arithmetic) higher rank subgroup, i.e. satisfying: A-rank(Γ) := ΣK ν -rank G(K ν ) 2, where the sum is taken over all those ν for which the projection of Γ to G(K ν ) is unbounded. Let H be any locally compact group and ϕ : Γ H a homomorphism. Assume further the following regularity condition: modulo the center of H, the subgroup Q < H generated by all the compact subgroups of H which are normalized by ϕ(γ) is tame. Then in each one of the following cases ϕ extends continuously to a homomorphism ϕ : G Γ H with closed image: 1. The homomorphism ϕ is not proper. 2. Denoting by H 0 the connected component of H, the normalizer of any compact open subgroup of the totally disconnected group H/H 0 is compact. For the ad hoc purpose of this conjecture, a topological group Q is called tame if its connected component Q 0 is a compact finite dimensional (i.e. Lie) group, and the group Q/Q 0 is a topologically finitely generated profinite (compact) group.
Prop. The CmSP (for Γ) is equivalent to the Conjecture (for Γ) when H is totally disconnected with no non-trivial compact normal subgroups, and Imϕ is dense. (Margulis NST is equivalent to the conjecture in the case where H is a discrete group with no non-trivial finite normal subgroups and ϕ is surjective.) For finite H the conjecture amounts to a sharp form of CSP: Every homo of Γ to a finite group extends to G Γ. It accounts for the various results on the cong kernel. The conjecture implies a stronger Margulis superrigidity: Every homo of Γ to GL n (k), k local field, extends to G Γ. Supported by additional results of Margulis (in his book). For H finite central extension of G(K ν ), the conjecture sharpens well known results of Deligne (78) on the non-residual finiteness of some central extensions of arithmetic groups.
Theorem. Fix n 3 and let π : G SL n (Q p ) be any non split central extension of topological groups. Then the lifted group Γ := π 1 (SL n (Z[ 1 p ])) < G is not virtually torsion (or center) free, nor is it residually finite. Proof. We show that every finite index subgroup Γ < Γ has center 1. Otherwise π : Γ π(γ ) < SL n (Z[ 1 p ]) is a group isomorphism, and can apply Conjecture (= previous Thm in this case) with Γ = π(γ ), H = G and ϕ = π 1 Γ In our case: G Γ /conn comp= closure of Γ in SL n (A f ) = SL n (Q p ) L for some compact group L. By non-splitness of the original extension, the restriction of any continuous extension ϕ to SL n (Q p ) must be trivial. Hence ϕ must have bounded image in H contradiction. The conjecture predicts that the splitness of π : SL n (R) SL n (R) over subgroups of SL n (Z) is a purely p-adic issue