Differentiability of quasi-conformal maps on the jungle gym

Gönye, Zsuzsanna

doi:10.1090/S0002-9947-06-04198-5

Differentiability of quasi-conformal maps on the jungle gym
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by Zsuzsanna Gönye PDF

Trans. Amer. Math. Soc. 359 (2007), 19-32 Request permission

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 $\mathbb {D}$ 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.

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