Ahlfors regular conformal dimension and Gromov-Hausdorff convergence

Abstract

We prove that the Ahlfors regular conformal dimension is upper semicontinuous with respect to Gromov-Hausdorff convergence when restricted to the class of uniformly perfect, uniformly quasi-selfsimilar metric spaces. A corollary is the upper semicontinuity of the Ahlfors regular conformal dimension of limit sets of discrete, quasiconvex-cocompact group of isometries of uniformly bounded codiameter of $\delta$-hyperbolic metric spaces under equivariant pointed Gromov-Hausdorff convergence of the spaces.Comment: Minor revision