arXiv math/01102952001-10-26math.DGmath.OA

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

Daniele Guido, Tommaso Isola

备注：: 17 pages, to appear on the Pacific Journal of Mathematics. This paper roughly corresponds to the third section of the unpublished math.DG/9809040
发表信息：: Pacific Journal of Mathematics, Volume 204, No. 1 (2002) 43-59

英文摘要：

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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