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Vincent Delecroix

Publications and source records attributed to Vincent Delecroix.

At least 19 recordsLinked to original sources
arXiv2026

Computing an e-net of a closed hyperbolic surface

Hyperbolic surfaces are a fundamental object in mathematics and play an increasingly important role in computational geometry and topology. A key ingredient in the design of efficient algorithms on such surfaces is the availability of a geometric discretization of controlled complexity. In this paper, we present the first algorithm for constructing e-nets on hyperbolic surfaces starting from a fundamental polygon representation. Our approach is based on Delaunay refinement and relies on maintaining Delaunay triangulations through edge flips. The size of an e-net cannot be bounded solely as a function of the genus because of the presence of arbitrarily long collars around short geodesics. To overcome this difficulty, we introduce the notion of a pseudo e-net, which decomposes the surface into e-thin cylinders together with a Delaunay triangulation over an e-net of the remaining thick part. As applications, we obtain algorithms for computing the length spectrum of an e-thick hyperbolic surface and for computing the systole from a pseudo log(sqrt(2))-net. These results demonstrate that Delaunay-based discretizations provide a practical and versatile framework for algorithmic computations on hyperbolic surfaces.

cs.CG↗
arXiv2026

On the size of k-irreducible triangulations

A triangulation of a surface is k-irreducible if every non-contractible curve has length at least k and any edge contraction breaks this property. Equivalently, every edge belongs to a non-contractible curve of length k and there are no shorter non-contractible curves. We prove that a k-irreducible triangulation of an orientable surface of genus g has $O(k^2g)$ triangles, which is optimal. This is an improvement over the previous best bound $k^{O(k)} g^2$ of Gao, Richter and Seymour [Journal of Combinatorial Theory, Series B, 1996].

cs.CG↗
arXiv2025

On the Computation of Schrijver's Kernels

The geometry of a graph $G$ embedded on a closed oriented surface $S$ can be probed by counting the intersections of $G$ with closed curves on $S$. Of special interest is the map $c \mapsto μ_G(c)$ counting the minimum number of intersections between $G$ and any curve freely homotopic to a given curve $c$. Schrijver [On the uniqueness of kernels, 1992] calls $G$ a kernel if for any proper graph minor $H$ of $G$ we have $μ_H < μ_G$. Hence, $G$ admits a minor $H$ which is a kernel and such that $μ_G = μ_H$. We show how to compute such a minor kernel of $G$ in $O(n^3 \log n)$ time where $n$ is the number of edges of $G$, and $g\ge 2$ is the genus of $S$. Our algorithm leverages a tight bound on the size of minimal bigons in a system of closed curves. It also relies on several subroutines of independent interest including the computation of the area enclosed by a curve and a test of simplicity for the lift of a curve in the universal covering of $S$. As a consequence of our minor kernel algorithm and a recent result of Dubois [Making multicurves cross minimally on surfaces, 2024], after a preprocessing that takes $O(n^3 \log n)$ time and $O(n)$ space, we are able to compute $μ_G(c)$ in $O(g (n + \ell) \log(n + \ell))$ time given any closed walk $c$ with $\ell$ edges. The state-of-the-art algorithm by Colin de Verdière and Erickson [Tightening non-simple paths and cycles on surfaces, 2010] would avoid constructing a kernel but would lead to a computation of $μ_G(c)$ in $O(g n \ell \log(n \ell))$ time (with a preprocessing that takes $O(gn\log n)$ time and $O(gn)$ space). Another consequence of the computation of minor kernels is the ability to decide in polynomial time whether two graph minors $H$ and $H'$ of $G$ satisfy $μ_H = μ_{H'}$.

cs.CG↗
arXiv2025

Reconfiguration of square-tiled surfaces

We consider a combinatorial reconfiguration problem on a subclass of quadrangulations of surfaces called square-tiled surfaces. Our elementary move is a shear in a cylinder that corresponds to a well-chosen sequence of diagonal flips that preserves the square-tiled properties. We conjecture that the connected components of this reconfiguration problem are in bijection with the connected components of the moduli space of quadratic differentials. We prove that the conjecture holds in the so-called hyperelliptic components of Abelian square-tiled surfaces. More precisely, we show that any two such square-tiled surfaces of genus $g$ can be connected by $O(g)$ powers of cylinder shears.

math.CO↗
arXiv2024

Diagonal flow detects topology of strata

We study the interplay between the diagonal flow on, and the topology of, a stratum component of a space of rooted quadratic differentials. We prove that the flow group -- the subgroup of the fundamental group generated by almost-flow loops -- equals the fundamental group. As a corollary, we show that the plus and minus modular Rauzy-Veech groups are finite-index subgroups of their ambient modular monodromy groups. This partially answers a question of Yoccoz. Using this, and recent advances on algebraic hulls and Zariski closures of symplectic monodromy groups, we prove that the Rauzy-Veech groups are Zariski dense in their ambient symplectic groups. Density, in turn, implies the simplicity of the plus and minus Lyapunov spectra of any component of any stratum of quadratic differentials. We thus establish the Kontsevich -- Zorich conjecture.

math.DS↗
arXiv2024

Infinite Translation Surfaces in the Wild

This book explores infinite-type translation surfaces and is intended as an introductory text for graduate and PhD students, as well as a reference for more advanced researchers. Chapter 1 introduces the three definitions of translation surfaces and meticulously proves their equivalence. It is enriched with numerous examples that are revisited throughout the book. Chapter 2 provides a detailed examination of the topological classification of infinite-type surfaces, the construction of infinite coverings of finite-type translation surfaces, and the structure of points within the metric completion. Chapter 3 investigates the affine symmetries of infinite-type translation surfaces, with special emphasis on infinite coverings of finite-type surfaces, the Hooper-Thurston-Veech construction, and affine homeomorphisms of finite-area infinite-type translation surfaces. Chapter 4 introduces infinite interval exchange transformations and employs them to demonstrate that the dynamics of translation flows are significantly more complex in the infinite-type context. The two appendices address hyperbolic geometry and the spectra of infinite graphs, respectively.

math.GT↗
arXiv2023

Higher genus meanders and Masur-Veech volumes

A meander can be seen as a pair of transversally intersecting simple closed curves on a 2-sphere. We consider pairs of transversally intersecting simple closed curves on a closed oriented surface of arbitrary genus g. The number of such higher genus meanders admits exponential upper and lower bounds as the number of intersections grows. Fixing the number n of bigons in the complement to the union of the two curves, we compute the precise asymptotics of genus g meanders with at most 2N intersections and show that this asymptotics is polynomial in N as N tends to infinity. We obtain a similar result for the number of positively intersecting pairs of oriented simple closed curves on a surface of genus g. We also compute the asymptotic probability of getting a meander from a random braid on a surface of genus g-1 with two boundary components. In order to effectively count meanders we identify them with integer points represented by certain square-tiled surfaces in the moduli spaces of Abelian and quadratic differentials and make use of recent advances in the geometry of these moduli spaces combined with asymptotic properties of Witten-Kontsevich 2-correlators on moduli spaces of complex curves.

math.GT↗
arXiv2023

Algorithms for Length Spectra of Combinatorial Tori

Consider a weighted, undirected graph cellularly embedded on a topological surface. The function assigning to each free homotopy class of closed curves the length of a shortest cycle within this homotopy class is called the marked length spectrum. The (unmarked) length spectrum is obtained by just listing the length values of the marked length spectrum in increasing order. In this paper, we describe algorithms for computing the (un)marked length spectra of graphs embedded on the torus. More specifically, we preprocess a weighted graph of complexity $n$ in time $O(n^2 \log \log n)$ so that, given a cycle with $\ell$ edges representing a free homotopy class, the length of a shortest homotopic cycle can be computed in $O(\ell+\log n)$ time. Moreover, given any positive integer $k$, the first $k$ values of its unmarked length spectrum can be computed in time $O(k \log n)$. Our algorithms are based on a correspondence between weighted graphs on the torus and polyhedral norms. In particular, we give a weight independent bound on the complexity of the unit ball of such norms. As an immediate consequence we can decide if two embedded weighted graphs have the same marked spectrum in polynomial time. We also consider the problem of comparing the unmarked spectra and provide a polynomial time algorithm in the unweighted case and a randomized polynomial time algorithm otherwise.

cs.CG↗
arXiv2022

Large genus asymptotic geometry of random square-tiled surfaces and of random multicurves

We study the combinatorial geometry of a random closed multicurve on a surface of large genus and of a random square-tiled surface of large genus. We prove that primitive components of a random multicurve represent linearly independent homology cycles with asymptotic probability 1 and that it is primitive with asymptotic probability $\sqrt{2}/2$. We prove analogous properties for random square-tiled surfaces. In particular, we show that all conical singularities of a random square-tiled surface belong to the same leaf of the horizontal foliation and to the same leaf of the vertical foliation with asymptotic probability 1. We show that the number of components of a random multicurve and the number of maximal horizontal cylinders of a random square-tiled surface of genus $g$ are both very well-approximated by the number of cycles of a random permutation for an explicit non-uniform measure on the symmetric group of $3g-3$ elements. In particular, we prove that the expected value of these quantities is asymptotically equivalent to $(\log(6g-6) + γ)/2 + \log 2$. These results are based on our formula for the Masur--Veech volume of the moduli space of holomorphic quadratic differentials combined with deep large genus asymptotic analysis of this formula performed by A.~Aggarwal and with the uniform asymptotic formula for intersection numbers of $ψ$-classes on the Deligne-Mumford compactification of the moduli space of curves proved by A.~Aggarwal.

math.GT↗
arXiv2022

Around the combinatorial unit ball of measured foliations on bordered surfaces

The volume $\mathscr{B}_Σ^{\rm comb}(\mathbb{G})$ of the unit ball -- with respect to the combinatorial length function $\ell_{\mathbb{G}}$ -- of the space of measured foliations on a stable bordered surface $Σ$ appears as the prefactor of the polynomial growth of the number of multicurves on $Σ$. We find the range of $s \in \mathbb{R}$ for which $(\mathscr{B}_Σ^{\rm comb})^{s}$, as a function over the combinatorial moduli spaces, is integrable with respect to the Kontsevich measure. The results depends on the topology of $Σ$, in contrast with the situation for hyperbolic surfaces where Arana-Herrera and Athreya (arXiv:1907.06287) recently proved an optimal square-integrability.

math.GT↗
arXiv2021

Kempe changes in degenerate graphs

We consider Kempe changes on the $k$-colorings of a graph on $n$ vertices. If the graph is $(k-1)$-degenerate, then all its $k$-colorings are equivalent up to Kempe changes. However, the sequence between two $k$-colorings that arises from the proof may be exponential in the number of vertices. An intriguing open question is whether it can be turned polynomial. We prove this to be possible under the stronger assumption that the graph has treewidth at most $k-1$. Namely, any two $k$-colorings are equivalent up to $O(kn^2)$ Kempe changes. We investigate other restrictions (list coloring, bounded maximum average degree, degree bounds). As a main result, we derive that given an $n$-vertex graph with maximum degree $Δ$, the $Δ$-colorings are all equivalent up to $O(n^2)$ Kempe changes, unless $Δ= 3$ and some connected component is a 3-prism.

math.CO↗
arXiv2020

Masur-Veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves

We express the Masur-Veech volume and the area Siegel-Veech constant of the moduli space $\mathcal{Q}_{g,n}$ of genus $g$ meromorphic quadratic differentials with $n$ simple poles as polynomials in the intersection numbers of $ψ$-classes with explicit rational coefficients. The formulae obtained in this article result from lattice point counts involving the Kontsevich volume polynomials that also appear in Mirzakhani's recursion for the Weil-Petersson volumes of the moduli spaces of bordered hyperbolic surfaces with geodesic boundaries. A similar formula for the Masur-Veech volume (though without explicit evaluation) was obtained earlier by Mirzakhani via completely different approach. Furthermore, we prove that the density of the mapping class group orbit of any simple closed multicurve $γ$ inside the ambient set of integral measured laminations computed by Mirzakhani coincides with the density of square-tiled surfaces having horizontal cylinder decomposition associated to $γ$ among all square-tiled surfaces in $\mathcal{Q}_{g,n}$. We study the resulting densities (or, equivalently, volume contributions) in more detail in the special case $n=0$. In particular, we compute the asymptotic frequencies of separating and non-separating simple closed geodesics on a closed hyperbolic surface of genus $g$ for small $g$ and we show that for large genera the separating closed geodesics are $\sqrt{\frac{2}{3πg}}\cdot\frac{1}{4^g}$ times less frequent.

math.GT↗
arXiv2020

Topological recursion for Masur-Veech volumes

We study the Masur-Veech volumes $MV_{g,n}$ of the principal stratum of the moduli space of quadratic differentials of unit area on curves of genus $g$ with $n$ punctures. We show that the volumes $MV_{g,n}$ are the constant terms of a family of polynomials in $n$ variables governed by the topological recursion/Virasoro constraints. This is equivalent to a formula giving these polynomials as a sum over stable graphs, and retrieves a result of \cite{Delecroix} proved by combinatorial arguments. Our method is different: it relies on the geometric recursion and its application to statistics of hyperbolic lengths of multicurves developed in \cite{GRpaper}. We also obtain an expression of the area Siegel--Veech constants in terms of hyperbolic geometry. The topological recursion allows numerical computations of Masur--Veech volumes, and thus of area Siegel--Veech constants, for low $g$ and $n$, which leads us to propose conjectural formulas for low $g$ but all $n$. We also relate our polynomials to the asymptotic counting of square-tiled surfaces with large boundaries.

math.GT↗
arXiv2020

Uniform Lower Bound for Intersection Numbers of $ψ$-Classes

We approximate intersection numbers $\big\langle ψ_1^{d_1}\cdots ψ_n^{d_n}\big\rangle_{g,n}$ on Deligne-Mumford's moduli space $\overline{\mathcal M}_{g,n}$ of genus $g$ stable complex curves with $n$ marked points by certain closed-form expressions in $d_1,\dots,d_n$. Conjecturally, these approximations become asymptotically exact uniformly in $d_i$ when $g\to\infty$ and $n$ remains bounded or grows slowly. In this note we prove a lower bound for the intersection numbers in terms of the above-mentioned approximatingexpressions multiplied by an explicit factor $λ(g,n)$, which tends to $1$ when $g\to\infty$ and $d_1+\dots+d_{n-2}=o(g)$.

math.GT↗
arXiv2020

admcycles -- a Sage package for calculations in the tautological ring of the moduli space of stable curves

The tautological ring of the moduli space of stable curves has been studied extensively in the last decades. We present a SageMath implementation of many core features of this ring. This includes lists of generators and their products, intersection numbers and verification of tautological relations. Maps between tautological rings induced by functoriality, that is pushforwards and pullbacks under gluing and forgetful maps, are implemented. Furthermore, many interesting cycle classes, such as the double ramification cycles, strata of k-differentials and hyperelliptic or bielliptic cycles are available. In this paper we show how to apply the package, including concrete example computations.

math.AG↗
arXiv2019

Approximation of the Lagrange and Markov spectra

The (classical) Lagrange spectrum is a closed subset of the positive real numbers defined in terms of diophantine approximation. Its structure is quite involved. This article describes a polynomial time algorithm to approximate it in Hausdorff distance. It also extends to approximate the Markov spectrum related to infimum of binary quadratic forms.

math.DS↗
arXiv2019

Coding Teichmüller flow using veering triangulations

We develop the theory of veering triangulations on oriented surfaces adapted to moduli spaces of half-translation surfaces. We use veering triangulations to give a coding of the Teichmüller flow on connected components of strata of quadratic differentials. We prove that this coding, given by a countable shift, has an approximate product structure and a roof function with exponential tails. This makes it conducive to the study of the dynamics of Teichmüller flow.

math.DS↗