Research Interests
My research interests lie in the intersection of geometry, topology, group theory and dynamics. Specifically, my work focusses on the areas of Teichmüller theory, mapping class groups, hyperbolic geometry and, more recently, the Lagrange and Markov spectra. 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 research on the Lagrange and Markov spectra - two complicated subsets of the real line related to Diophantine approximation - is focused on understanding their intricate structure in a region that has remained difficult to investigate.
My CV is available here.
Papers and Preprints
- On the monodromy and spin parity of single-cylinder origamis in the minimal stratum, with Tarik Aougab, Adam Friedman-Brown, and Jason Ma. In preparation.
- Minimal constructions of single-cylinder pillowcase covers. In preparation.
- On the classical Lagrange and Markov spectra: new results on the local dimension, and the geometry of the difference set, with Harold Erazo and Carlos Gustavo Moreira. Preprint, 51 pages. arXiv:2405.20581
Abstract:
Let L and M denote the classical Lagrange and Markov spectra, respectively. It is known that L⊂M and that M∖L≠∅. Inspired by three questions asked by the third author in previous work investigating the fractal geometric properties of the Lagrange and Markov spectra, we investigate the function dloc(t) that gives the local Hausdorff dimension at a point t of L′. Specifically, we construct several intervals (having non-trivial intersection with L′) on which dloc(t) is non-decreasing. We also prove that the respective intersections of M′ and M′′ with these intervals coincide. Furthermore, we completely characterize the local dimension of both spectra when restricted to those intervals. Finally, we demonstrate the largest known elements of the difference set M∖L and describe two new maximal gaps of M nearby.
- Meanders, hyperelliptic pillowcase covers, and the Johnson filtration. Submitted, 26 pages. arXiv:2210.11332
Abstract:
We provide minimal constructions of meanders with particular combinatorics. Using these meanders, we give minimal constructions of hyperelliptic pillowcase covers with a single horizontal cylinder and simultaneously a single vertical cylinder so that one or both of the core curves are separating curves on the underlying surface. In the case where both of the core curves are separating, we use these surfaces in a construction of Aougab-Taylor in order to prove that for any hyperelliptic connected component of the moduli space of quadratic differentials with no poles there exist ratio-optimising pseudo-Anosovs lying arbitrarily deep in the Johnson filtration and stabilising the Teichmüller disk of a quadratic differential lying in this connected component.
Slides:
The slides for a talk I gave at the Weihnachtsworkshop on Geometry and Number Theory 2022 at the Universität des Saarlandes, Germany.
- New gaps on the Lagrange and Markov spectra, with Carlos Matheus and Carlos Gustavo Moreira. To appear in J. Théor. Nombres Bordeaux, 21 pages. arXiv:2209.12876
Abstract:
Let L and M denote the Lagrange and Markov spectra, respectively. It is known that L⊂M and that M∖L≠∅. In this work, we exhibit new gaps of L and M using two methods. First, we derive such gaps by describing a new portion of M∖L near to 3.938: this region (together with three other candidates) was found by investigating the pictures of L recently produced by V. Delecroix and the last two authors with the aid of an algorithm explained in one of the appendices to this paper. As a by-product, we also get the largest known elements of M∖L and we improve upon a lower bound on the Hausdorff dimension of M∖L obtained by the last two authors together with M. Pollicott and P. Vytnova (heuristically, we get a new lower bound of 0.593 on the dimension of M∖L). Secondly, we use a renormalisation idea and a thickness criterion (reminiscent from the third author's PhD thesis) to detect infinitely many maximal gaps of M accumulating to Freiman's gap preceding the so-called Hall's ray [4.52782956616...,∞)⊂L.
- Non-planarity of SL(2,ℤ)-orbits of origamis in ℋ(2), with Carlos Matheus. Bull. London Math. Soc. doi: 10.1112/blms.12849
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.
Poster:
A poster I presented at a visit of the EPSRC's senior team to the University of Bristol.
- 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.
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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.
- 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.
Abstract:
Let L and M denote the classical Lagrange and Markov spectra, respectively. It is known that L⊂M and that M∖L≠∅. Inspired by three questions asked by the third author in previous work investigating the fractal geometric properties of the Lagrange and Markov spectra, we investigate the function dloc(t) that gives the local Hausdorff dimension at a point t of L′. Specifically, we construct several intervals (having non-trivial intersection with L′) on which dloc(t) is non-decreasing. We also prove that the respective intersections of M′ and M′′ with these intervals coincide. Furthermore, we completely characterize the local dimension of both spectra when restricted to those intervals. Finally, we demonstrate the largest known elements of the difference set M∖L and describe two new maximal gaps of M nearby.
Abstract:
We provide minimal constructions of meanders with particular combinatorics. Using these meanders, we give minimal constructions of hyperelliptic pillowcase covers with a single horizontal cylinder and simultaneously a single vertical cylinder so that one or both of the core curves are separating curves on the underlying surface. In the case where both of the core curves are separating, we use these surfaces in a construction of Aougab-Taylor in order to prove that for any hyperelliptic connected component of the moduli space of quadratic differentials with no poles there exist ratio-optimising pseudo-Anosovs lying arbitrarily deep in the Johnson filtration and stabilising the Teichmüller disk of a quadratic differential lying in this connected component.
Slides:
The slides for a talk I gave at the Weihnachtsworkshop on Geometry and Number Theory 2022 at the Universität des Saarlandes, Germany.
Abstract:
Let L and M denote the Lagrange and Markov spectra, respectively. It is known that L⊂M and that M∖L≠∅. In this work, we exhibit new gaps of L and M using two methods. First, we derive such gaps by describing a new portion of M∖L near to 3.938: this region (together with three other candidates) was found by investigating the pictures of L recently produced by V. Delecroix and the last two authors with the aid of an algorithm explained in one of the appendices to this paper. As a by-product, we also get the largest known elements of M∖L and we improve upon a lower bound on the Hausdorff dimension of M∖L obtained by the last two authors together with M. Pollicott and P. Vytnova (heuristically, we get a new lower bound of 0.593 on the dimension of M∖L). Secondly, we use a renormalisation idea and a thickness criterion (reminiscent from the third author's PhD thesis) to detect infinitely many maximal gaps of M accumulating to Freiman's gap preceding the so-called Hall's ray [4.52782956616...,∞)⊂L.
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.
Poster:
A poster I presented at a visit of the EPSRC's senior team to the University of Bristol.
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.
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.
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.
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.