  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 previously
(math.OA/9802015, cf. also math.DG/0110294). 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.
