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

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