Lipschitz Cohomology, Novikov Conjecture, and Expanders

We present sufficient conditions for the cohomology of a closed aspherical manifold to be proper Lipschitz in the sense of Connes–Gromov–Moscovici. 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 by Carlsson and Pedersen. 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 examples of exotic groups.