On Carrasco Piaggio's theorem characterizing quasisymmetric maps from compact doubling spaces to Ahlfors regular spaces

TL;DR

This paper analyzes Carrasco Piaggio's theorem linking Ahlfors regular conformal gauges to hyperbolic fillings, simplifying the construction and examining how specific properties influence the induced metric.

Contribution

It provides a simplified hyperbolic filling construction and investigates the impact of theorem's properties on the metric, clarifying the theorem's components.

Findings

Simplified hyperbolic filling construction

Identification of property effects on the induced metric

Enhanced understanding of Carrasco Piaggio's theorem

Abstract

In this note we deconstruct and explore the components of a theorem of Carrasco Piaggio, which relates Ahlfors regular conformal gauge of a compact doubling metric space to weights on Gromov-hyperbolic fillings of the metric space. We consider a construction of hyperbolic filling that is simpler than the one considered by Carrasco Piaggio, and we determine the effect of each of the four properties postulated by Carrasco Piaggio on the induced metric on the compact metric space.