Nonarithmetic superrigid groups: Counterexamples to Platonov’s conjecture

Margulis showed that “most” arithmetic groups are superrigid. Platonov conjectured, conversely, that finitely generated linear groups which are superrigid must be of “arithmetic type.” We construct counterexamples to Platonov’s Conjecture. 1. Platonov’s conjecture that rigid linear groups are aritmetic (1.1) Representation rigid groups. Let Γ be a finitely generated group. By a representation of Γ we mean a finite dimensional complex representation, i.e. essentially a homomorphism ρ : Γ −→ GLn(C), for some n. We call Γ linear if some such ρ is faithful (i.e. injective). We call Γ representation rigid if, in each dimension n ≥ 1, Γ admits only finitely many isomorphism classes of simple (i.e. irreducible) representations. Platonov ([P-R, p. 437]) conjectured that if Γ is representation rigid and linear then Γ is of “arithmetic type” (see (1.2)(3) below). Our purpose here is to construct counterexamples to this conjecture. In fact our counterexamples are representation superrigid, in the sense that the Hochschild-Mostow completion A(Γ) is finite dimensional (cf. [BLMM] or [L-M]). The above terminology is justified as follows (cf. [L-M]): If Γ = 〈s1, . . . , sd〉 is given with d generators, then the map ρ 7→ (ρ(s1), . . . , ρ(sd)) identifies Rn(Γ) = Hom(Γ,GLn(C)) with a subset of GLn(C) d. In fact Rn(Γ) is easily seen to be an affine subvariety. It is invariant under the simultaneous conjugation action of GLn(C) on GLn(C) d. The algebraic-geometric quotient Xn(Γ) = GLn(C)\\Rn(Γ) exactly parametrizes the isomorphism classes of semi-simple n-dimensional representations of Γ. It is sometimes called the n-dimensional “character variety” of Γ. With this terminology we see that Γ is representation rigid if and only if all character varieties of Γ are finite (or zero-dimensional). In other words, there are no moduli for simple Γ-representations. Work partially supported by the US-Israel Binational Science Foundation. 1152 HYMAN BASS AND ALEXANDER LUBOTZKY (1.2) Examples and remarks. (1) If Γ ≤ Γ is a subgroup of finite index then Γ is representation rigid if and only if Γ is representation rigid (cf. [BLMM]). Call groups Γ and Γ1 (abstractly) commensurable if they have finite index subgroups Γ ≤ Γ and Γ′1 ≤ Γ1 which are isomorphic. In this case Γ is representation rigid if and only if Γ1 is so. (2) Let K be a finite field extension of Q, S a finite set of places containing all archimedean places, and K(S) the ring of S-integers in K. Let G be a linear algebraic group over K, and G(K(S)) the group of S-integral points in G(K). Under certain general conditions, for semi-simple G (see (2.1) below), the Margulis superrigidity theorem applies here, and it implies in particular that G(K(S)) is representation rigid. (3) Call a group Γ of “arithmetic type” if Γ is commensurable (as in (1)) with a product n ∏