Representative functions on discrete solvable groups

Abstract

The topic here is the representation of discrete groups as automorphisms of finite-dimensional vector spaces over a field. The results here are mainly generalizations to the class of solvable groups of finite torsion-free rank of results of Mostow concerning polycyclic groups. In discussing the finite-dimensional representations of a group f over a field k, one is led to consider the “continuous dual” k[r]O of the group algebra, where “continuous” refers to the topology on k[r] in which a fundamental system of neighborhoods of zero is the family of kernels of finitedimensional representations of r over k. k[r]” has the structure of a coalgebra--indeed, of a Hopf algebra-and the locally finite-dimensional k[f]-modules correspond to the k[f]’ comodules. The advantage of looking at k[f]” is that any finite-dimensional k[F]“-comodule is a subcomodule of the direct sum of finitely many copies of k[r]“. A further advantage is that, when k is algebraically closed, k[fJ” is the polynomial algebra of a pro-affme algebraic group Gk(IJ whose rational representations correspond exactly to the locally finite-dimensional representations of f over k. This brings to bear the structural results on pro-afftne algebraic groups, which are strongest when the base field k is of characteristic zero. In Section 1, we examine the unipotent radical U,(T) of the pro-affine group Gk(IJ associated with a discrete group r, and we prove that, on the category of solvable groups of finite torsion-free rank (i.e., the groups of type A, in Mal’cev’s classification [lo] ), the assignment of Uk(ZJ to f is an exact functor to the category of unipotent affrne algebraic groups. In Section 2, we consider a certain homomorphic image Bk(I) of G,(T) which we call the basic k-group associated with I-; this group is the “lowest” homomorphic image of Gk(r) to “preserve” the unipotent radical U,(f). The assignment of Bk(f) is functorial on a subcategory of the category of 272 OO21-8693/87 $3.00