On the conformal gauge of a compact metric space

Matias Carrasco Piaggio (1)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités

In this article we study the Ahlfors regular conformal gauge of a compact metric space $(X,d)$, and its conformal dimension $\mathrm{dim}_{AR}(X,d)$. Using a sequence of finite coverings of $(X,d)$, we construct distances in its Ahlfors regular conformal gauge of controlled Hausdorff dimension. We obtain in this way a combinatorial description, up to bi-Lipschitz homeomorphisms, of all the metrics in the gauge. We show how to compute $\mathrm{dim}_{AR}(X,d)$ using the critical exponent $Q_N$ associated to the combinatorial modulus.

Mots clés: Ahlfors regular; conformal gauge; conformal dimension; combinatorial modulus; Gromov-hyperbolic.

Mots clés: Gromov-hyperbolic

Domaines: Géométrie métrique [math.MG]

Matias Carrasco Piaggio. On the conformal gauge of a compact metric space. 2012. ⟨hal-00687840v2⟩