Conformal Assoaud dimension as the critical exponent for combinatorial modulus

Abstract

The conformal Assouad dimension is the infimum of all possible values of Assouad dimension after a quasisymmetric change of metric. We show that the conformal Assouad dimension equals a critical exponent associated to the combinatorial modulus for any compact doubling metric space. This generalizes a similar result obtained by Carrasco Piaggio for the Ahlfors regular conformal dimension to a larger family of spaces. We also show that the value of conformal Assouad dimension is unaffected if we replace quasisymmetry with power quasisymmetry in its definition.