Aspects of Quasiconformal Homogeneity

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New Zealand Journal of Mathematics

Vol. 39, (2009), Pages 117-132


Jianhua Gong

Department of Mathematical Science

United Arab Emirates University

P.O. Box 17551

Al Ain

United Arab Emirates

mailto:j.gong@uaeu.ac.ae

Gaven Martin

Institute for Advanced Study

Massey University

Private Bag

Auckland

New Zealand

mailto:g.j.martin@massey.ac.nz




Abstract This paper studies the notion of quasiconformal homogeneity in various different settings - relative with respect to domains and ambiently. For instance we give an example of a compact set E which is homogeneous with respect to a quasiconformal family of maps of a domain containing E, but which is not quasiconformally homogeneous with respect to \mathbb{R}^n. We construct Cantor sets which are K--quasiconformally homogeneous for each K > 1 and explicitly identify their Hausdorff dimension, among other things.

Keywords quasiconformal, homogeneity, quasiconformal groups, Lie group.

Classification (MSC2000) Primary: 30C60.

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