O'Shea, S; (2013) Stably free modules over virtually free groups. Doctoral thesis , UCL (University College London).

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Abstract

We study stably free modules over various group rings Z[G], using the method of Milnor patching. In particular, we construct innite sets of stably free modules of rank one over various rings. Let Fn denote the free group on n generators. The two classes of group rings under consideration are: (i) Z[GxFn], where G is nite nilpotent and of non square-free order, and n > 2; (ii) Z[Q(12m) x C1], where Q(12m) is the binary polyhedral group of order 12m. The modules in question are constructed as pullbacks arising from bre square decompositions of the group rings. We also study the D(2)-problem of low-dimensional topology. We give an affirmative answer to the D(2)-problem for the dihedral group of order 4n, assuming the group ring Z[D4n] satisfies torsion free cancellation. By results of Swan, Endo, and Miyata, this happens for a number of small primes n. Johnson has shown that the groups D4n+2 satisfy the D(2)-property, but his result relies on the fact that D4n+2 has periodic cohomology, a property not shared by D4n. This forces us to introduce the torsion free cancellation hypothesis, and to explicitly realize the group of k-invariants (Z/4n).

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