Cyclic Cohomology and Higher Rank Lattices

Abstract

We give a new proof of the absence of non-trivial idempotents in the group ring of torsion-free cocompact lattices in SL(n,C). It is based on the following procedure. We lift the class of the trace in the cyclic cohomology of the group ring to the crossed product of the smooth functions on the Furstenberg boundary of SL(n,C) with the lattice. We then perform a Dirac-dual Dirac method on smooth algebras in analytic cyclic cohomology. This is based on a form of equivariant Bott periodicity under compact Lie groups in analytic cyclic cohomology. We make crucial use of the Baum-Connes conjecture for solvable Lie groups. There is also a chapter in which we prove that the class of the unit in the K-theory of the crossed product of the continuous functions on the visibility boundary of the symmetric space of a real semisimple Lie group with torsion-free discrete subgroups of that Lie group is not torsion if the lattice is not cocompact. In case the lattice is cocompact, we show that the class of unit is torsion if and only if the rank of the Lie group is the same as that of a maximal compact subgroup.