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A Vector Field Approach to Mapping Class Actions
Author(s) -
Gardiner F. P.,
Lakic N.
Publication year - 2006
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611505015558
Subject(s) - mathematics , cantor set , mapping class group , pure mathematics , class (philosophy) , complement (music) , mathematics subject classification , vector field , homotopy , type (biology) , bounded function , space (punctuation) , surface (topology) , mathematical analysis , geometry , artificial intelligence , chemistry , ecology , biochemistry , complementation , biology , computer science , gene , phenotype , linguistics , philosophy
We present a vector field method for showing that certain subgroups of the mapping class group Γ 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 Ω of a Cantor set in C. When the Cantor set has bounded geometric type, we show that Γ(Ω) acts on the Teichmüller space T (Ω) properly discontinuously. Also, we apply the same method to show that the pure mapping class group Γ 0 Ω ∪ ∞ acts properly discontinuously on T Ω ∪ ∞. 2000 Mathematics Subject Classification 30F60 (primary), 32G15, 30C70, 30C75 (secondary).