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

Abstract The goal of this paper is to provide an expository description of a result of Carrasco Piaggio [8] connecting the Ahlfors regular conformal dimension of a compact uniformly perfect doubling metric space with the combinatorial $p$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> -moduli of the metric space. We give detailed construction of a metric associated with the $p$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> -modulus of the space when the $p$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> -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 in [10,6].