2001/10/26 by Daniele Guido, Tommaso Isola
Mathematics · #math.DG #math.OA #msc:55M10 #msc:46L87 #msc:58B34
published as Pacific Journal of Mathematics, Volume 204, No. 1 (2002) 43-59 · 17 pages, to appear on the Pacific Journal of Mathematics. This paper roughly corresponds to the third section of the unpublished math.DG/9809040
arxiv created 2001/10/26 · arxiv updated 2009/11/30
A nonnegative number dinfinity, 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 alpha0 defined previously (math.OA/9802015, cf. also math.DG/0110294). Thus the dimensional interpretation of alpha0 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.