Metric Spaces of Arbitrary Finitely-Generated Scaling Group

TL;DR

This paper constructs specific metric spaces with prescribed finitely-generated scaling groups, expanding understanding of how groups of scaling factors relate to geometric structures.

Contribution

It demonstrates that any finitely generated subgroup of positive reals can be realized as the scaling group of a bi-Lipschitz equivalent graph-based metric space.

Findings

Constructed spaces have the desired scaling groups

Spaces are bi-Lipschitz equivalent to finite degree graphs

Shows the diversity of scaling groups in metric space theory

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 $\Gamma$.