Kamali, P.; (2010) Stably free modules over in�nite group algebras. Doctoral thesis , UCL (University College London).

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Abstract

We study �finitely generated stably-free modules over infinite integral group algebras by using the language of cyclic algebras and relating it to well-known results in K-theory. For G a free or free abelian group and Q8n, the quaternionic group of order 8n, we show that there exist infinitely many isomorphically distinct stably-free modules of rank 1 over the integral group algebra of the group \Gamma = Q8n x G whenever n admits an odd divisor. This result implies that the stable class of the augmentation ideal \Omega{_1}Z displays infi�nite splitting at minimal level whenever G is the free abelian group on at least 2 generators. This is of relevance to low dimensional topology, in particular when computing homotopy modules of a cell complex with fundamental group \Gamma.

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