Ubiquity of ratio optimisers

Luke Jeffreys (Glasgow)

Monday 8th October, 2018 16:00-17:00 Maths 311B

Abstract

The systole map from Teichmüller space to the curve complex is a coarse map that associates to a marked hyperbolic surface the shortest closed geodesic in the hyperbolic metric.  Almost twenty years ago, Masur-Minsky proved the breakthrough result that the systole map is coarsely Lipschitz.  More recently, Gadre-Hironaka-Kent-Leininger showed that the optimal Lipschitz constant is 1/log g, where g is the genus of the surface.  A key part of their work was the construction of certain pseudo-Anosov maps known as ratio optimisers.  Subsequently, Aougab-Taylor gave infinitely many examples of ratio optimisers using filling pairs of curves and a construction due to Thurston.  We explore the flexibility of their construction to show that ratio optimisers are ubiquitous: they are found in every connected component of every stratum of Abelian differentials.

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