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

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

2001-10-26

arxiv.org/abs/math/0110295v1

Pacific Journal of Mathematics, Volume 204, No. 1 (2002) 43-59

Mathematics

Differential Geometry

17 pages, to appear on the Pacific Journal of Mathematics. This paper roughly corresponds to the third section of the unpublis

Scientific paper

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.

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