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

Daniele Guido and Tommaso Isola

Vol. 204 (2002), No. 1, 43–59

DOI: 10.2140/pjm.2002.204.43

Abstract

A nonnegative number d_∞, 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 α₀ defined in a previous paper [D. Guido, T. Isola, J. Funct. Analysis, 176 (2000)]. Thus the dimensional interpretation of α₀ 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.

Milestones

Received: 10 April 2000

Revised: 16 October 2000

Published: 1 May 2002