Geometric and analytic quasiconformality in metric measure spaces

Abstract

We prove the equivalence between geometric and analytic definitions of quasiconformality for a homeomorphism $f:X\to Y$ between arbitrary locally finite separable metric measure spaces, assuming no metric hypotheses on either space. When $X$ and $Y$ have locally $Q$-bounded geometry and $Y$ is contained in an Alexandrov space of curvature bounded above, the sharpness of our results implies that, as in the classical case, the modular and pointwise outer dilatations of $\map$ are related by $K_O(f)=\mathrm{esssup}{H}_O(x,f)$.

15 pages. Flaw in covering argument for Theorem 1.2 corrected. Flaws in Remarks 4.3 and 4.4 corrected. Accepted for publication by the Proceedings of the American Mathematical Society