Conformal Assouad dimension and modulus
by
Stephen Keith
Australian National University
Coauthors: Tomi Laakso (University of Helsinki)

The existence of families of curves with non-vanishing modulus in a metric measure space is a strong and useful property, perhaps most appreciated in the study of abstract quasiconformal and quasisymmetric mappings, Sobolev spaces, Poincaré inequalities, and geometric rigidity questions. In this talk (which will be my second talk of the conference), I will explain a recent result by myself and T. Laakso concerning the existence of families of curves with non-vanishing modulus in weak tangents of certain metric measure spaces, and show that this existence is fundamentally related to the conformal Assouad dimension of the underlying metric space. I will then discuss applications, including a recent result of Bonk and Kleiner which states the if a Gromov hyperbolic group G has boundary quasisymmetrically homeomorphic to an Ahlfors Q-regular metric 2-sphere with conformal Assouad dimension Q, then G acts discretely, cocompactly, and isometrically on hyperbolic 3-space.

Date received: December 11, 2003