arXiv research

Compact groups can be represented as dessin automorphisms.

The Loch Ness Monster admits many regular dessins d'enfants and different holomorphic structures.

Recently, Gareth Jones observed that every finite group $G$ can be realized as the group of automorphisms of some dessin d'enfant ${\mathcal D}$. In this paper, complementing Gareth's result, we prove that for every possible action of $G$ as a group of orientation-preserving homeomorphisms on a closed orientable surfac…

We show how to define a canonical Riemannian metric on a "dessin d'enfants' drawn on a topological surface. This gives a possible explanation of a claim of A. Grothendieck.

Machine learning classifies complex geometric patterns with high accuracy.

Study minima of geodesic lengths for specific curves on surfaces.

A cumbersome hypothesis for Viro patchworking of real algebraic curves is the convexity of the given subdivision. It is an open question in general to know whether the convexity is necessary. In the case of trigonal curves we interpret Viro method in terms of dessins d'enfants. Gluing the dessins d'enfants in a coheren…

We study topological recursion on the irregular spectral curve $xy^2-xy+1=0$, which produces a weighted count of dessins d'enfant. This analysis is then applied to topological recursion on the spectral curve $xy^2=1$, which takes the place of the Airy curve $x=y^2$ to describe asymptotic behaviour of enumerative proble…

Prove realizability of genus-0 branch data for tetrahedral coverings of the sphere.

The Hurwitz problem asks which ramification data are realizable, that is appear as the ramification type of a covering. We use dessins d'enfant to show that families of genus 1 regular ramification data with small changes are realizable with the exception of four families which were recently shown to be nonrealizable. …

Paper studies metric ribbon graphs and provides a recursion for their volumes.

We give a combinatorial description of closed curves on oriented surfaces in terms of certain permutations, called charts. We describe automorphisms of curves in terms of charts and compute the total number of curves counted with appropriate weights. We also discuss relations between curves, Grothendieck dessins d'enfa…

We consider triangulations of surfaces with edges painted three colors so that edges of each triangle have different colors. Such structures arise as Belyi data (or Grothendieck dessins d'enfant), on the other hand they enumerate pairs of permutations determined up to a common conjugation. The topic of these notes is l…

The geometry of cosets in the subgroups H of the two-generator free group G =\textless{} a, b \textgreater{} nicely fits, via Grothendieck's dessins d'enfants, the geometry of commutation for quantum observables. Dessins stabilize point-line incidence geometries that reflect the commutation of (generalized) Pauli opera…

Oriented ribbon graphs (dessins d'enfant) are graphs embedded in oriented surfaces. The Bollobás-Riordan-Tutte polynomial is a three-variable polynomial that extends the Tutte polynomial to oriented ribbon graphs. A quasi-tree of a ribbon graph is a spanning subgraph with one face, which is described by an ordered chor…

Characterizes Lamé equations with finite monodromy on flat tori.

We compute the number of (weak) equivalence classes of branched covers from a surface of genus g to the sphere, with 3 branching points, degree 2k, and local degrees over the branching points of the form (2,...,2), (2h+1,1,2,...,2), (d_1,...,d_m), for several values of g and h. We obtain explicit formulae of arithmetic…

We complete the topological classification of real algebraic non-singular curves of bidegree $(5, 5)$ on the quadric ellipsoid. We show in particular that previously known restrictions form a complete system for this bidegree. Therefore, the main part of the paper concerns the construction of real algebraic curves. Our…

Oriented ribbon graphs (dessins d'enfant) are graphs embedded in oriented surfaces. A quasi-tree of a ribbon graph is a spanning subgraph with one face, which is described by an ordered chord diagram. We show that for any link diagram $L$, there is an associated ribbon graph whose quasi-trees correspond bijectively to …

We continue our computation, using a combinatorial method based on Gronthendieck's dessins d'enfant, of the number of (weak) equivalence classes of surface branched covers matching certain specific branch data. In this note we concentrate on data with the surface of genus g as source surface, the sphere as target surfa…

We investigate modular embeddings for semi-arithmetic Fuchsian groups. First we prove some purely algebro-geometric or even topological criteria for a regular map from a smooth complex curve to a quaternionic Shimura variety to be covered by a modular embedding. Then we set up an adelic formalism for modular embeddings…

New approach finds minima of geodesic lengths for non-uniform fillings.

To a branched cover f between orientable surfaces one can associate a certain branch datum D(f), that encodes the combinatorics of the cover. This D(f) satisfies a compatibility condition called the Riemann-Hurwitz relation. The old but still partly unsolved Hurwitz problem asks whether for a given abstract compatible …

For the existence of a branched covering Sigma~ --> Sigma between closed surfaces there are easy necessary conditions in terms of chi(Sigma~), chi(Sigma), orientability, the total degree, and the local degrees at the branching points. A classical problem dating back to Hurwitz asks whether these conditions are also suf…

A dessin is a 2-cell embedding of a connected bipartite graph into an orientable closed surface. An automorphism of a dessin is a permutation of the edges of the underlying graph which preserves the colouring of the vertices and extends to an orientation-preserving self-homeomorphism of the supporting surface. A dessin…

Harer and Zagier proved a recursion to enumerate gluings of a $2d$-gon that result in an orientable genus $g$ surface, in their work on Euler characteristics of moduli spaces of curves. Analogous results have been discovered for other enumerative problems, so it is natural to pose the following question: how large is t…

Grothendieck's Esquisse d'un programme is often referred to for the ideas it contains on dessins d'enfants, the Teichm{ü}ller tower, and the actions of the absolute Galois group on these objects or their etale fundamental groups. But this program contains several other important ideas. In particular, motivated by surfa…

We review some ideas of Grothendieck and others on actions of the absolute Galois group Γ Q of Q (the automorphism group of the tower of finite extensions of Q), related to the geometry and topology of surfaces (mapping class groups, Teichm{ü}ller spaces and moduli spaces of Riemann surfaces). Grothendieck's motivation…

A classical result states that the determinant of an alternating link is equal to the number of spanning trees in a checkerboard graph of an alternating connected projection of the link. We generalize this result to show that the determinant is the alternating sum of the number of quasi-trees of genus j of the dessin o…

Abstract: Study of geometric structures on surfaces using various tools.

We consider a rather special class of translation surfaces (called M-Origamis in this work) that are obtained from dessins by a construction introduced by Martin Möller. We give a new proof with a more combinatorial flavour of Möller's theorem that $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ acts faithfully on the…