Cyclic homology and the unit conjecture.
by
Krzysztof Pawlowicz
Warsaw University

The talk will concern an application of the cyclic homology to the investigation of the unit conjecture. The unit conjecture states that the group algebra kG of a torsion-free group G over a field k has only 'obvious' units i.e., ones of the form ag with a in k, g in G. The idea of engaging the cyclic homology is motivated and inspired by the succes of the method in the investigation of the idempotent conjecture. With any idempotent of the algebra R one can associate a sequence of elements in HC_*(A). If for A we take the group algebra kG and G is for example polycyclic-by-finite then the groups HC_n(A) vanish for n large enough. This observation suffices to prove non-existence of idempotents in kG different from 0 and 1. One can devise a similar construction involving units of kG.

Date received: August 10, 2000