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Differentiability of quasi-conformal maps on the jungle gym

Author: Zsuzsanna Gönye
Journal: Trans. Amer. Math. Soc. 359 (2007), 19-32
MSC (2000): Primary 30C62, 28A78; Secondary 30F35
Published electronically: August 15, 2006
MathSciNet review: 2247880
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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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Additional Information

Zsuzsanna Gönye
Affiliation: Department of Mathematics, Polytechnic University, Brooklyn, New York 11201

Keywords: Fuchsian group, Kleinian group, quasi-conformal map, jungle gym, Hausdorff dimension
Received by editor(s): September 9, 2004
Published electronically: August 15, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.