Differentiability of quasi-conformal maps on the jungle gym

Abstract

We obtain a result on the quasi-conformal self-maps of jungle gyms, a divergence-type group. If the dilatation is compactly supported, then the induced map on the boundary of the covering disc D is differentiable with non-zero derivative on a set of Hausdorff dimension 1. As one of the corollaries, we show that there are quasi-symmetric homeomorphisms over divergence-type groups such that for all sets E the Hausdorff dimension of E and f(E c ) cannot both be less than 1. This shows an important difference.between finitely generated and divergence-type groups.