Luke Jeffreys

Research Interests

My research interests lie in the areas of Teichmüller theory, mapping class groups, and hyperbolic geometry. I am currently investigating how the coarse geometry of Teichmüller space relates to certain properties of quadratic differentials and to other aspects of Teichmüller dynamics. I am also interested in the connections between hyperbolic geometry and the large-scale geometry of Teichmüller space.

My CV is available here.

Papers and Preprints

  1. Statistical hyperbolicity for harmonic measure, with Aitor Azemar and Vaibhav Gadre. Submitted, 13 pages. arXiv:1909.13811

    Abstract: We consider harmonic measures that arise from random walks on the mapping class group determined by a probability distribution that has finite first moment with respect to the Teichmüller metric, and whose support generates a non-elementary subgroup. We prove that Teichmüller space with the Teichmüller metric is statistically hyperbolic for such a harmonic measure.

  2. Single-cylinder square-tiled surfaces and the ubiquity of ratio-optimising pseudo-Anosovs. Submitted, 39 pages. arXiv:1906.02016

    Abstract: In every connected component of every stratum of Abelian differentials, we construct square-tiled surfaces with one vertical and one horizontal cylinder. We show that for all but the hyperelliptic components this can be achieved in the minimum number of squares necessary for a square-tiled surface in that stratum. For the hyperelliptic components, we show that the number of squares required is strictly greater and construct surfaces realising these bounds.

    Using these surfaces, we demonstrate that pseudo-Anosov homeomorphisms optimising the ratio of Teichmüller to curve graph translation length are, in a reasonable sense, ubiquitous in the connected components of strata of Abelian differentials. Finally, we present a further application to filling pairs on punctured surfaces by constructing filling pairs whose algebraic and geometric intersection numbers are equal.

  3. Minimally intersecting filling pairs on the punctured surface of genus two. Topology Appl. 254 (2019) 101-106.
    doi: 10.1016/j.topol.2018.12.011

    Abstract: In this short note, we construct a minimally intersecting pair of simple closed curves that fill a genus 2 surface with an odd, at least 3, number of punctures. This finishes the determination of minimally intersecting filling pairs for all surfaces completing the work of Aougab-Huang and Aougab-Taylor.