Posts Tagged ‘flat surfaces’

Erwan Lanneau gave a talk in Orsay about his work on surfaces of translation. Pseudo-Anosov homeomorphisms are homeomorphisms of such surfaces which are affine away from the singular set of the surface and whose differential is hyperbolic. The dilation factor is the dominant eigenvalue of that differential. It is a Perron number.

The minimum of the dilation factor for given genus is known for geni 1 and 2 only (the techniques behind could be extended to genus 3 but not farther). One also knows that c_1/g\leq \log \delta(g)\leq c_2/g.

The main result of the talk is that the above does not hold when restricted to a given type of translation surfaces. More precisely, the moduli space of translation surfaces of given genus splits into connected components (at most three), one of them corresponding to hyperellipticity and the following holds:

Theorem (Boissy-Lanneau) Let \Phi be a pseudo-Anosov on a hyperelliptic translation surface of genus g admitting an involution with 2g+2 fixed points). Assume that \Phi has a unique singularity. Then its dilation is strictly greater than \sqrt{2} (but approach this value as g\to\infty).

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