# Aspects of Quasiconformal Homogeneity

### 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

Gaven Martin

Massey University

Private Bag

Auckland

New Zealand

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.