Abstract
In this note, we show that given a closed, orientable genus-g surface S g , any hyperbolic toral automorphism has a positive power which induces a quadratic, orientable pseudo-Anosov homeomorphism on S g . 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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Brown, R. Generating Quadratic Pseudo-Anosov Homeomorphisms of Closed Surfaces. Geometriae Dedicata 97, 129–150 (2003). https://doi.org/10.1023/A:1023636817155
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DOI: https://doi.org/10.1023/A:1023636817155