An asymptotic dimension for metric spaces, and the 0-th Novikov-Shubin invariant

A nonnegative number d(infinity), called asymptotic dimension, is associated with any metric space. Such number detects the asymptotic properties of the space (being zero on bounded metric spaces), fulfills the properties of a dimension, and is invariant under rough isometries. It is then shown that for a class of open manifolds with bounded geometry the asymptotic dimension coincides with the 0-th Novikov-Shubin number alpha(0) defined in a previous paper [D. Guido, T. Isola, J. Funct. Analysis, 176 (2000)]. Thus the dimensional interpretation of alpha(0) given in the mentioned paper in the framework of noncommutative geometry is established on metrics grounds. Since the asymptotic dimension of a covering manifold coincides with the polynomial growth of its covering group, the stated equality generalises to open manifolds a result by Varopoulos.