Groups of units of integral group rings commensurable with direct products of free-by-free groups

Cohomology Rings of Almost-direct Products of Free Groups

An almost-direct product of free groups is an iterated semidirect product of finitely generated free groups in which the action of the constituent free groups on the homology of one another is trivial. We determine the structure of the cohomology ring of such a group. This is used to analyze the topological complexity of the associated Eilenberg-Mac Lane space. 1. Almost direct products of free...

Involutions and Free Pairs of Bicyclic Units in Integral Group Rings of Non-nilpotent Groups

If ∗ : G → G is an involution on the finite group G, then ∗ extends to an involution on the integral group ring Z[G]. In this paper, we consider whether bicyclic units u ∈ Z[G] exist with the property that the group 〈u, u∗〉, generated by u and u∗, is free on the two generators. If this occurs, we say that (u, u∗) is a free bicyclic pair. It turns out that the existence of u depends strongly upo...

Bass Units as Free Factors in Integral Group Rings of Simple Groups

Let G be a finite group, u a Bass unit based on an element a of G of prime order, and assume that u has infinite order modulo the center of the units of the integral group ring ZG. It was recently proved that if G is solvable then there is a Bass unit or a bicyclic unit v and a positive integer n such that the group generated by un and vn is a non-abelian free group. It has been conjectured tha...

Groups generated by two bicyclic units in integral group rings

Central Units of Integral Group Rings of Nilpotent Groups

In this paper a finite set of generators is given for a subgroup of finite index in the group of central units of the integral group ring of a finitely generated nilpotent group. In this paper we construct explicitly a finite set of generators for a subgroup of finite index in the centre Z(U(ZG)) of the unit group U(ZG) of the integral group ring ZG of a finitely generated nilpotent group G. Ri...