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arXiv · 2107.01972

Polynomial growth and asymptotic dimension

Panos Papasoglu
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Abstract

Bonamy et al \cite{BBEGLPS} showed that graphs of polynomial growth have finite asymptotic dimension. We refine their result showing that a graph of polynomial growth strictly less than $n^{k+1}$ has asymptotic dimension at most $k$. As a corollary Riemannian manifolds of bounded geometry and polynomial growth strictly less than $n^{k+1}$ have asymptotic dimension at most $k$. We show also that there are graphs of growth $ 0$ and infinite asymptotic Assouad-Nagata dimension.

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Panos Papasoglu. 2021-07-05. Polynomial growth and asymptotic dimension. https://arxiv.org/abs/2107.01972

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