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A VECTOR FIELD APPROACH TO MAPPING CLASS ACTIONS

Published online by Cambridge University Press:  20 February 2006

F. P. GARDINER  and
N. LAKIC
F. P. GARDINER
Affiliation:
Department of Mathematics, Brooklyn College, CUNY, 2900 Bedford Avenue, Brooklyn, NY 11210, USAfgardiner@gc.cuny.edu
N. LAKIC
Affiliation:
Department of Mathematics, Lehman College, CUNY, 250 Bedford Park Boulevard West, Bronx, NY 10468, USAnlakic@lehman.cuny.edu
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Abstract

We present a vector field method for showing that certain subgroups of the mapping class group $\Gamma$ of a Riemann surface of infinite topological type act properly discontinuously. We apply the method to the group of homotopy classes of quasiconformal self-maps of the complement $\Omega$ of a Cantor set in $\mathbb{C}$. When the Cantor set has bounded geometric type, we show that $\Gamma(\Omega)$ acts on the Teichmüller space $T(\Omega)$ properly discontinuously. Also, we apply the same method to show that the pure mapping class group $\Gamma_0(\Omega \cup \{\infty\})$ acts properly discontinuously on $T(\Omega \cup \{\infty\})$.

Keywords

30F60 (primary)32G1530C7030C75 (secondary)
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 92 , Issue 2 , March 2006 , pp. 403 - 427
Copyright
2006 London Mathematical Society

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