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

DOI10.2140/PJM.2002.204.43zbMATH Open1053.58017arXivmath/0110295OpenAlexW2020200631MaRDI QIDQ701245

Daniele Guido, Tommaso Isola

Publication date: 22 October 2002

Published in: Pacific Journal of Mathematics (Search for Journal in Brave)

Abstract: 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.

Full work available at URL: https://arxiv.org/abs/math/0110295

Mathematics Subject Classification ID

Invariance and symmetry properties for PDEs on manifolds (58J70) Other ``noncommutative mathematics based on (C^*)-algebra theory (46L89) Noncommutative global analysis, noncommutative residues (58J42)

Related Items (2)

The problem of completeness for Gromov-Hausdorff metrics on \(C^{\ast}\)-algebras ⋮ An intermediate quasi-isometric invariant between subexponential asymptotic dimension growth and Yu's property A

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