Quasiconformal motions and isomorphisms of continuous families of Möbius groups

In their paper [17], Sullivan and Thurston introduced the notion of quasiconformal motions, and proved an extension theorem for quasiconformal motions over an interval. We prove some new properties of (normalized) quasiconformal motions of a closed set E in the Riemann sphere, over connected Hausdor...

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Bibliographic Details
Published in:Israel journal of mathematics 2012-03, Vol.188 (1), p.177-194
Main Authors: Jiang, Yunping, Mitra, Sudeb, Shiga, Hiroshige
Format: Article
Language:English
Subjects:
Algebra
Analysis
Applications of Mathematics
Group Theory and Generalizations
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Theoretical
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Summary:In their paper [17], Sullivan and Thurston introduced the notion of quasiconformal motions, and proved an extension theorem for quasiconformal motions over an interval. We prove some new properties of (normalized) quasiconformal motions of a closed set E in the Riemann sphere, over connected Hausdorff spaces. As a spin-off, we strengthen the result of Sullivan and Thurston, and show that if a quasiconformal motion of E over an interval has a certain group-equivariance property, then the extended quasiconformal motion can be chosen to have the same group-equivariance property. Our main theorem proves a result on isomorphisms of continuous families of Möbius groups arising from a group-equivariant quasiconformal motion of E over a path-connected Hausdorff space. Our techniques connect the Teichmüller space of the closed set E with quasiconformal motions of E .
ISSN:0021-2172
1565-8511
DOI:10.1007/s11856-011-0098-1