arXiv subjects

Luke Jeffreys

Publications and source records attributed to Luke Jeffreys.

10 recordsLinked to original sources

arXiv2026

Euler characteristics of $SL(2,\mathbb{Z})$-orbit graphs of origamis

The $SL(2,\mathbb{Z})$-orbits of primitive $n$-squared origamis can be represented by finite four-regular graphs. It is a conjecture of McMullen that the orbit graphs of such origamis in the stratum $\mathcal{H}(2)$ form an expander family. 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 primitive origamis in the Prym loci of $\mathcal{H}(4)$ and $\mathcal{H}(6)$. Assuming conjectures of Zmiaikou and Delecroix--Leli\`evre, we establish the same in $\mathcal{H}(1,1)$ and for two families of non-Prym origamis in $\mathcal{H}(4)$. Finally, assuming a stronger conjecture concerning orbit growth in low-complexity strata, we establish that for any family of $SL(2,\mathbb{Z})$-orbit graphs of primitive origamis in a stratum with one or two conical singularities, the absolute values of the Euler characteristics of the graphs go to infinity with the number of squares along a density one subsequence. By relating the genus of the elliptic generator orbit graphs to the genus of the associated arithmetic Teichm\"uller curves, we also recover results of Mukamel and extend results of Torres-Teigel--Zachhuber establishing that the genus of these Teichm\"uller curves in $\mathcal{H}(2)$ and the Prym loci of $\mathcal{H}(4)$ and $\mathcal{H}(6)$ 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 Teichm\"uller curves, and on counts of pseudo-Anosov diffeomorphisms with bounded dilatation.

math.GT↗

arXiv2025

Diameter bounds for $SL(2,\mathbb{Z})$-orbits of origamis in $\mathcal{H}(2)$ and the Prym loci in $\mathcal{H}(4)$ and $\mathcal{H}(6)$

Using algorithms implicit in the classification of $SL(2,\mathbb{Z})$-orbits of primitive origamis in the stratum $\mathcal{H}(2)$ due to Hubert-Leli\`evre and McMullen, we give diameter bounds on the resulting orbit graphs. Since the machinery of McMullen from $\mathcal{H}(2)$ is generalised and reused in Lanneau and Nguyen's classification of the orbits of Prym eigenforms in $\mathcal{H}(4)$ and $\mathcal{H}(6),$ 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 $O(N^{2/3}\log N)$, where $N$ is the size of the orbit graph.

math.GT↗

arXiv2025

On the monodromy and spin parity of single-cylinder origamis in the minimal stratum

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., that of Eskin-Kontsevich-Zorich, Filip-Forni-Matheus, Guti\'{e}rrez-Romo, Kany-Matheus, Matheus-Yoccoz-Zmiaikou, and 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.

math.GT↗

arXiv2024

On the classical Lagrange and Markov spectra: new results on the local dimension and the geometry of the difference set

Let $L$ and $M$ denote the classical Lagrange and Markov spectra, respectively. It is known that $L\subset M$ and that $M\setminus L\neq\varnothing$. 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 $d_{loc}(t)$ 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 $d_{loc}$ 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 $M\setminus L$ and describe two new maximal gaps of $M$ nearby.

math.NT↗

arXiv2022

Meanders, hyperelliptic pillowcase covers, and the Johnson filtration

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 Teichm\"uller disk of a quadratic differential lying in this connected component.

math.GT↗

arXiv2022

New gaps on the Lagrange and Markov spectra

Let $L$ and $M$ denote the Lagrange and Markov spectra, respectively. It is known that $L\subset M$ and that $M\setminus L\neq\varnothing$. 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 $M\setminus L$ 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 $M\setminus L$ and we improve upon a lower bound on the Hausdorff dimension of $M\setminus L$ 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 $M\setminus L$). 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 $[4.52782956616...,\infty)\subset L$.

math.NT↗

arXiv2021

Non-planarity of $\text{SL}(2,\mathbb{Z})$-orbits of origamis in $\mathcal{H}(2)$

We consider the $\text{SL}(2,\mathbb{Z})$-orbits of primitive $n$-squared origamis in the stratum $\mathcal{H}(2)$. In particular, we consider the 4-valent graphs obtained from the action of $\text{SL}(2,\mathbb{Z})$ 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 $K_{3,3}$ minor, where $K_{3,3}$ is the complete bipartite graph on two sets of three vertices.

math.GT↗

arXiv2019

Statistical hyperbolicity for harmonic measure

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 Teichmuller metric, and whose supports generate non-elementary subgroups. We prove that Teichmuller space with the Teichmuller metric is statistically hyperbolic for such a harmonic measure.

math.GT↗

arXiv2019

Single-cylinder square-tiled surfaces and the ubiquity of ratio-optimising pseudo-Anosovs

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 Teichm\"uller 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.

math.GT↗

arXiv2018

Minimally intersecting filling pairs on the punctured surface of genus two

In this short note, we construct a minimally intersecting pair of simple closed curves that fill a genus 2 surface with an odd, greater than 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.

math.GT↗