On Carrasco Piaggio's theorem connecting combinatorial modulus and Ahlfors regular conformal dimension

TL;DR

This paper explains Carrasco Piaggio's theorem linking the Ahlfors regular conformal dimension of certain metric spaces with their combinatorial p-moduli, using hyperbolic filling techniques.

Contribution

It provides an expository account and detailed construction of a metric related to the p-modulus, connecting conformal dimension with combinatorial modulus.

Findings

Established a construction of metrics in the conformal gauge based on p-modulus.

Clarified the relationship between Ahlfors regular conformal dimension and combinatorial p-moduli.

Utilized hyperbolic filling tools to facilitate the construction and analysis.

Abstract

The goal of this paper is to provide an expository description of a result of Carrasco Piaggio connecting the Ahlfors regular conformal dimension of a compact uniformly perfect doubling metric space with the combinatorial $p$-moduli of the metric space. We give detailed construction of a metric associated with the $p$-modulus of the space when the $p$-modulus is zero, so that the constructed metric is in the Ahlfors regular conformal gauge of the metric space. To do so, we utilize the tools of hyperbolic filling, developed first by Gromov and by Bourdon and Pajot.