Lipschitz Cohomology, Novikov conjecture, and Expanders

Abstract

We present sufficient conditions for the cohomology of a closed aspherical manifold to be proper Lipschitz in sense of Connes-Gromov-Moscovici [CGM]. The conditions are stated in terms of the Stone-Čech compactification of the universal cover of a manifold. We show that these conditions are formally weaker than the sufficient conditions for the Novikov conjecture given in [CP]. Also we show that the Cayley graph of the fundamental group of a closed aspherical manifold with proper Lipschitz cohomology cannot contain an expander in the coarse sense. In particular, this rules out a Lipschitz cohomology approach to the Novikov Conjecture for recent Gromov's examples of exotic groups.