Luke Jeffreys

Research Interests

My research interests lie in the intersection of geometry, topology and group theory. Specifically, my work focusses on the areas of Teichmüller theory, mapping class groups, and hyperbolic geometry. I am particularly interested in constructions of flat surfaces, the study of their combinatorics and geometry, and more generally the geometry and dynamics of Teichmüller space and related spaces.

My CV is available here.

Papers and Preprints

  1. Non-planarity of SL(2,Z)-orbits of origamis in H(2), with Carlos Matheus. Submitted, 9 pages. arXiv:2107.08786
    Abstract:
    We consider the SL(2,Z)-orbits of primitive n-squared origamis in the stratum H(2). In particular, we consider the 4-valent graphs obtained from the action of SL(2,Z) with respect to a generating set of size two. We prove that, apart from the orbit for n = 3 and one of the orbits for n = 5, all of the obtained graphs are non-planar. Specifically, in each of the graphs we exhibit a K3,3 minor, where K3,3 is the complete bipartite graph on two sets of three vertices.

  2. Statistical hyperbolicity for harmonic measure, with Aitor Azemar and Vaibhav Gadre. Int. Math. Res. Not. 2022, no. 8, 6289–6309. doi: 10.1093/imrn/rnaa277
    Abstract:
    We consider harmonic measures that arise from random walks on the mapping class group determined by probability distributions that have finite first moment with respect to the Teichmüller metric, and whose supports generate non-elementary subgroups. We prove that Teichmüller space with the Teichmüller metric is statistically hyperbolic for such a harmonic measure.

  3. Single-cylinder square-tiled surfaces and the ubiquity of ratio-optimising pseudo-Anosovs. Trans. Amer. Math. Soc. 374 (2021) 5739-5781. doi: 10.1090/tran/8374
    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.

  4. 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.

PhD Thesis (Awarded the University of Glasgow William Jack Prize)

Single-cylinder square-tiled surfaces: Constructions and applications, University of Glasgow, July 2020

LMS Newsletter Microthesis

Single-Cylinder Square-Tiled Surfaces, LMS Newsletter, Issue: 491 - November 2020, pp. 36-37