RELATIVE COMPLETIONS OF LINEAR GROUPS OVER Z [ t ] AND Z [ t , t − 1 ]

We compute the completion of the groups SLn(Z[t]) and SLn(Z[t, t ]) relative to the obvious homomorphisms to SLn(Q); this is a generalization of the classical Malcev completion. We also make partial computations of the rational second cohomology of these groups. The Malcev (or Q-) completion of a group Γ is a prounipotent group P defined over Q together with a homomorphism φ : Γ → P satisfying the following universal mapping property: If ψ : Γ → U is a map of Γ into a prounipotent group then there is a unique map Φ : P → U such that ψ = Φφ. If H1(Γ,Q) = 0, then the group P is trivial and is therefore useless for studying Γ. In particular, the Malcev completions of the groups SLn(Z[t]) and SLn(Z[t, t −1]) are trivial when n ≥ 3 (this follows from the work of Suslin [14]). Here we consider Deligne’s notion of relative completion. Suppose ρ : Γ → S is a representation of Γ in a semisimple linear algebraic group over Q. Suppose that the image of ρ is Zariski dense in S. The completion of Γ relative to ρ is a proalgebraic group G over Q, which is an extension of S by a prounipotent group U , and homomorphism ρ̃ : Γ → G which lifts ρ and has Zariski dense image. When S is the trivial group, G is simply the classical Malcev completion. The relative completion satisfies an obvious universal mapping property. The basic theory of relative completion was developed by R. Hain [5] (and independently by E. Looijenga (unpublished)), and is reviewed in Section 2 below. In this paper we consider the completions of the groups SLn(Z[t]) and SLn(Z[t, t −1]) relative to the homomorphisms to SLn(Q) given by setting t = 0 (respectively, t = 1). There is an obvious candidate for the relative completion, namely the proalgebraic group SLn(Q[[T ]]). The map SLn(Z[t]) → SLn(Q[[T ]]) is the obvious inclusion and the map SLn(Z[t, t ]) → SLn(Q[[T ]]) Received by the editors February 8, 2008. 1991 Mathematics Subject Classification. Primary 55P60, 20G35, 20H05; Secondary 20G10, 20F14. Supported by an NSF Postdoctoral Fellowship, grant no. DMS-9627503. c ©1998 American Mathematical Society