Generating Quadratic Pseudo-Anosov Homeomorphisms of Closed Surfaces |
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Authors: | Richard Brown |
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Affiliation: | (1) Department of Mathematics and Statistics, The American University, 4400 Massachusetts Ave., NW, Washington, DC, 20016-8050, U.S.A. |
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Abstract: | In this note, we show that given a closed, orientable genus-g surface Sg, any hyperbolic toral automorphism has a positive power which induces a quadratic, orientable pseudo-Anosov homeomorphism on Sg. To show this, we lift Anosov toral automorphisms through a ramified topological covering and present the lifted homeomorphism via a standard set of Lickorish twists. This construction provides a general method of producing pseudo-Anosov maps of closed surfaces with predetermined orientable foliations and quadratic dilatation. Since these lifted automorphisms have orientable foliations, this construction is a sort of converse to that of Franks and Rykken [Trans. Amer. Math. Soc. 1999], who established that one can associate to a quadratic pseudo-Anosov homeomorphism with oriented unstable foliation a hyperbolic toral automorphism. |
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Keywords: | pseudo-Anosov homeomorphism mapping class group Dehn twist surface group |
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