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arXiv · 2205.04367

Metric Spaces of Arbitrary Finitely-Generated Scaling Group

Daniel N. Levitin
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Abstract

For a metric space $X$ with a compatible measure $\mu$, Genevois and Tessera defined the Scaling Group of $(X,\mu)$ as the subgroup $\Gamma$ of $\mathbb{R}_{>0}$ of positive real numbers $\gamma$ for which there are quasi-isometries of $X$ coarsely scaling $\mu$ by a factor of $\gamma$. We show that for any finitely generated subgroup $\Gamma$ of $\mathbb{R}_{>0}$ there exists a space $N_\Gamma$, bi-Lipschitz equivalent to a graph of finite degree, with scaling group $\Gamma$.

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Daniel N. Levitin. 2022-05-09. Metric Spaces of Arbitrary Finitely-Generated Scaling Group. https://doi.org/10.1512/iumj.2024.73.9918

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