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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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