Property A as metric amenability and its applications to geometry

Abstract

Property A was introduced by Guoliang Yu as a metric version of a well-known group invariant, amenability. The new notion turned out to be extremely useful in several areas of mathematics. We prove an averaging theorem for Property A and explore its applications. The first is a construction of metric spaces which do not have Property A but admit a coarse embedding into a Hilbert space. This gives an answer to an open problem in coarse geometry and disproves a conjecture due to A.N.Dranishnikov. The second is a connection between type of asymptotic dimension and isoperimetric profiles, which allows to answer an open question of J.Roe. The third application is a proof of the zero-in-the-spectrum conjecture on certain Galois covers of compact manifolds.