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
- Minimal constructions of single-cylinder pillowcase covers. In preparation.
- Euler characteristics of SL(2,ℤ)-orbit graphs of origamis, with Carlos Matheus. Submitted, 40 pages. arXiv:2602.21984
Abstract:
The SL(2,ℤ)-orbits of primitive n-squared origamis can be represented by finite four-regular graphs. It is a conjecture of McMullen that the family of orbit graphs of such origamis in the stratum ℋ(2) is an expander family. In this work, we provide indirect evidence for this conjecture by proving that the absolute values of the Euler characteristics of the graphs in this family go to infinity with the number of squares n. This generalises previous work of the authors, in which we established eventual non-planarity for this family, and provides the strongest indirect evidence to date for McMullen's conjecture.
We also prove that the same phenomenon holds for the SL(2,ℤ)-orbits of primitive origamis in the Prym loci of ℋ(4) and ℋ(6) that have been classified by Lanneau–Nguyen.
Assuming conjectures of Zmiaikou and Delecroix–Lelièvre concerning SL(2,ℤ)-orbits in ℋ(1,1) and ℋ(4), respectively, we establish the same result for all origamis in ℋ(1,1) and for two families of non-Prym origamis in ℋ(4).
Finally, assuming a stronger conjecture concerning orbit growth in low-complexity strata, we prove an analogous result in greater generality. That is, we establish that for any family of SL(2,ℤ)-orbit graphs of primitive n-squared origamis in a stratum of translation surfaces with one or two conical singularities, the absolute values of the Euler characteristics of the graphs typically go to infinity with the number of squares in the sense that this holds along a density one subsequence.
By relating the genus of the graphs constructed with elliptic generators to the genus of the associated arithmetic Teichmüller curves, we are also able to recover results of Mukamel and extend results of Torres-Teigel–Zachhuber establishing that the genera of these Teichmüller curves in ℋ(2) and the Prym loci of ℋ(4) and ℋ(6) also go to infinity.
The proofs rely on counts of integral points on algebraic hypersurfaces using methods of Bombieri–Pila and Browning–Gorodnik, on counts of orbifold points on Teichmüller curves (via CM points on Hilbert modular surfaces in some cases), and on counts of pseudo-Anosov diffeomorphisms with bounded dilatation.
- Diameter bounds for SL(2,ℤ)-orbits of origamis in ℋ(2) and the Prym loci in ℋ(4) and ℋ(6), with Carlos Matheus. 36 pages, to appear in Math. Z.
Abstract:
Using algorithms implicit in the classification of SL(2,ℤ)-orbits of primitive origamis in the stratum ℋ(2) due to Hubert-Lelièvre and McMullen, we give diameter bounds on the resulting orbit graphs. Since the machinery of McMullen from ℋ(2) is generalised and reused in Lanneau and Nguyen's classification of the orbits of Prym eigenforms in ℋ(4) and ℋ(6), we are also able to obtain diameter bounds for the orbit graphs in this setting as well. In each stratum, we obtain diameter bounds of the form O(N2/3log N), where N is the size of the orbit graph.
- On the monodromy and spin parity of single-cylinder origamis in the minimal stratum, with Tarik Aougab, Adam Friedman-Brown, and Jiajie Ma. Submitted, 56 pages. arXiv:2502.09498
Abstract:
In a paper with Menasco–Nieland, the first author constructed factorially
many origamis in the minimal stratum of the moduli space of translation surfaces having
simultaneously a single vertical cylinder and a single horizontal cylinder. Moreover, these
origamis were constructed using the minimal number of squares required for origamis in
the minimal stratum. We shall call such origamis minimal [1,1]-origamis.
In this work, we calculate all of the spin parities of the Aougab–Menasco–Nieland
origamis, and we therefore determine the connected component of the minimal stratum
within which each is contained. Motivated by understanding the SL(2, ℤ)-orbits of these
origamis, we investigate their monodromy groups, in particular proving that all of them
are alternating or projective special linear groups. In fact, we prove more generally that
the monodromy group of a minimal [1,1]-origami must almost always be a finite simple
group. Finally, we determine the Kontsevich–Zorich monodromies of these origamis in
low genus and give a conjecture in general.
Note that previous works in the literature (e.g., Eskin–Kontsevich–Zorich, Filip–Forni–Matheus, Gutiérrez-Romo, Kany–Matheus, Matheus–Yoccoz–Zmiaikou, Zmiaikou, Zorich) often chose to discuss just one of these SL(2, ℤ)-invariants at a time: in particular, to the best our knowledge, this is one of the first places where all of these SL(2, ℤ)-invariants
are computed explicitly in a single paper for such a large family of origamis.
Slides:
The slides for a talk I gave at the Glasgow 2024 meeting of the North British Geometric Group Theory Seminar (NBGGT).
- 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. Acta Arith. 220 (2025), 29-100. doi: 10.4064/aa240821-26-5
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.
Slides:
The slides for a talk I gave at the UC San Diego Group Actions Seminar.
- Meanders, hyperelliptic pillowcase covers, and the Johnson filtration. Geom. Dedicata 218, 93 (2024). doi: 10.1007/s10711-024-00936-w
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. J. Théor. Nombres Bordeaux. 36 (2024), no. 1, 311-338. doi: 10.5802/jtnb.1280
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.
Slides:
The slides for a talk I gave at the Selected Topics in Mathematics - Online Edition seminar at the University of Liverpool.
- Non-planarity of SL(2,ℤ)-orbits of origamis in ℋ(2), with Carlos Matheus. Bull. London Math. Soc. 55 (2023), 2258-2269. 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.
-
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:
The SL(2,ℤ)-orbits of primitive n-squared origamis can be represented by finite four-regular graphs. It is a conjecture of McMullen that the family of orbit graphs of such origamis in the stratum ℋ(2) is an expander family. In this work, we provide indirect evidence for this conjecture by proving that the absolute values of the Euler characteristics of the graphs in this family go to infinity with the number of squares n. This generalises previous work of the authors, in which we established eventual non-planarity for this family, and provides the strongest indirect evidence to date for McMullen's conjecture.
We also prove that the same phenomenon holds for the SL(2,ℤ)-orbits of primitive origamis in the Prym loci of ℋ(4) and ℋ(6) that have been classified by Lanneau–Nguyen.
Assuming conjectures of Zmiaikou and Delecroix–Lelièvre concerning SL(2,ℤ)-orbits in ℋ(1,1) and ℋ(4), respectively, we establish the same result for all origamis in ℋ(1,1) and for two families of non-Prym origamis in ℋ(4).
Finally, assuming a stronger conjecture concerning orbit growth in low-complexity strata, we prove an analogous result in greater generality. That is, we establish that for any family of SL(2,ℤ)-orbit graphs of primitive n-squared origamis in a stratum of translation surfaces with one or two conical singularities, the absolute values of the Euler characteristics of the graphs typically go to infinity with the number of squares in the sense that this holds along a density one subsequence.
By relating the genus of the graphs constructed with elliptic generators to the genus of the associated arithmetic Teichmüller curves, we are also able to recover results of Mukamel and extend results of Torres-Teigel–Zachhuber establishing that the genera of these Teichmüller curves in ℋ(2) and the Prym loci of ℋ(4) and ℋ(6) also go to infinity.
The proofs rely on counts of integral points on algebraic hypersurfaces using methods of Bombieri–Pila and Browning–Gorodnik, on counts of orbifold points on Teichmüller curves (via CM points on Hilbert modular surfaces in some cases), and on counts of pseudo-Anosov diffeomorphisms with bounded dilatation.Abstract:
Using algorithms implicit in the classification of SL(2,ℤ)-orbits of primitive origamis in the stratum ℋ(2) due to Hubert-Lelièvre and McMullen, we give diameter bounds on the resulting orbit graphs. Since the machinery of McMullen from ℋ(2) is generalised and reused in Lanneau and Nguyen's classification of the orbits of Prym eigenforms in ℋ(4) and ℋ(6), we are also able to obtain diameter bounds for the orbit graphs in this setting as well. In each stratum, we obtain diameter bounds of the form O(N2/3log N), where N is the size of the orbit graph.
Abstract:
In a paper with Menasco–Nieland, the first author constructed factorially
many origamis in the minimal stratum of the moduli space of translation surfaces having
simultaneously a single vertical cylinder and a single horizontal cylinder. Moreover, these
origamis were constructed using the minimal number of squares required for origamis in
the minimal stratum. We shall call such origamis minimal [1,1]-origamis.
In this work, we calculate all of the spin parities of the Aougab–Menasco–Nieland origamis, and we therefore determine the connected component of the minimal stratum within which each is contained. Motivated by understanding the SL(2, ℤ)-orbits of these origamis, we investigate their monodromy groups, in particular proving that all of them are alternating or projective special linear groups. In fact, we prove more generally that the monodromy group of a minimal [1,1]-origami must almost always be a finite simple group. Finally, we determine the Kontsevich–Zorich monodromies of these origamis in low genus and give a conjecture in general.
Note that previous works in the literature (e.g., Eskin–Kontsevich–Zorich, Filip–Forni–Matheus, Gutiérrez-Romo, Kany–Matheus, Matheus–Yoccoz–Zmiaikou, Zmiaikou, Zorich) often chose to discuss just one of these SL(2, ℤ)-invariants at a time: in particular, to the best our knowledge, this is one of the first places where all of these SL(2, ℤ)-invariants are computed explicitly in a single paper for such a large family of origamis.Slides:
The slides for a talk I gave at the Glasgow 2024 meeting of the North British Geometric Group Theory Seminar (NBGGT).
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.
Slides:
The slides for a talk I gave at the UC San Diego Group Actions Seminar.
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.
Slides:
The slides for a talk I gave at the Selected Topics in Mathematics - Online Edition seminar at the University of Liverpool.
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.