Ahlfors regular conformal dimension and Gromov-Hausdorff convergence

TL;DR

This paper investigates the behavior of the Ahlfors regular conformal dimension under Gromov-Hausdorff convergence, establishing upper semicontinuity and continuity results for specific classes of hyperbolic metric spaces and their limit sets.

Contribution

It proves upper semicontinuity of the Ahlfors regular conformal dimension in certain metric spaces and shows its continuity for limit sets of specific hyperbolic groups under Gromov-Hausdorff convergence.

Findings

Ahlfors regular conformal dimension is upper semicontinuous in uniformly perfect, quasi-selfsimilar spaces.

Continuity of the conformal dimension is established for limit sets of hyperbolic groups.

Results apply to spaces with bounded codiameter under equivariant 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. Moreover we show the continuity of the Ahlfors regular conformal dimension in case 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.