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Polynomial growth and asymptotic dimension

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Abstract

Bonamy et al. [4] showed that graphs of polynomial growth have finite asymptotic dimension. We refine their result showing that a graph of polynomial growth strictly less than nk+1 has asymptotic dimension at most k. As a corollary Riemannian manifolds of bounded geometry and polynomial growth strictly less than nk+1 have asymptotic dimension at most k.

We show also that there are graphs of growth < n1+ϵ for any ϵ > 0 and infinite asymptotic Assouad—Nagata dimension.

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Authors and Affiliations

  1. Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter (550) Woodstock Road, Oxford, OX2 6GG, UK

    Panos Papasoglu

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  1. Panos Papasoglu
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Correspondence to Panos Papasoglu.

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Papasoglu, P. Polynomial growth and asymptotic dimension. Isr. J. Math. 255, 985–1000 (2023). https://doi.org/10.1007/s11856-023-2479-7

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  • DOI: https://doi.org/10.1007/s11856-023-2479-7

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