NEW INVARIANTS FOR GROUPS

Abstract

The aim of this paper is to begin the development of a theory of higher-dimensional Alexander ideals for groups. The classical Alexander ideals of a finitely generate group take the form of an ascending chain of ideals in a commutative ring. For a group of type FPn, we define a group invariant in every dimension up to n, also in the form of a chain of ideals, the ideals in dimension 1 being precisely the chain of Alexander ideals. We show how these invariants can be calculated for certain groups, such as R. Thompson's group, (relatively) aspherical groups and graphs of groups, and we demonstrate that the higher-dimension invariants can distinguish groups which the classical Alexander ideals cannot.We also define Alexander-type ideals for modules as well as a number of secondary group invariants in the form of polynomials and 'ranks', which are connected with work of Swan, Hattori and Stallings, K. S. Brown and Lustig. Finally, we explain how this theory can be extended to monoids.