Related papers: Hecke triangle Groups and Dessin d'enfant

Each finite and connected bipartite graph induces a finite collection of non-isomorphic dessins d'enfants, that is, $2$-cell embeddings of it into some closed orientable surface. We describe an algorithm to compute all these dessins…

We point out an explicit connection between graphs drawn on compact Riemann surfaces defined over the field $\bar{\mathbb{Q}}$ of algebraic numbers --- so-called Grothendieck's {\it dessins d'enfants} --- and a wealth of distinguished…

Grothendieck's dessins d'enfants arise with ever-increasing frequency in many areas of 21st century mathematical physics. In this paper, we review the connections between dessins and the theory of Hecke groups. Focussing on the restricted…

Mixed graphs can be seen as digraphs with arcs and edges (or digons, that is, two opposite arcs). In this paper, we consider the case where such graphs are bipartite and in which the undirected and directed degrees are one. The best graphs,…

We show that the absolute Galois group acts faithfully on the set of Hubbard trees. Hubbard trees are finite planar trees, equipped with self-maps, which classify postcritically finite polynomials as holomorphic dynamical systems on the…

Dessins d'enfants are combinatorial structures on compact Riemann surfaces defined over algebraic number fields, and regular dessins are the most symmetric of them. If G is a finite group, there are only finitely many regular dessins with…

In this work, we derive the low index subgroups of the extended Hecke, Hecke and the Picard groups using tools in color symmetry theory. We also present the low index subgroups of the modular group.

Let $G$ be a finite group and let $H_1,H_2<G$ be two subgroups. In this paper, we are concerned with the bipartite graph whose vertices are $G/H_1\cup G/H_2$ and a coset $g_1H_1$ is connected with another coset $g_2H_2$ if and only if…

For a finite group $G$, we define the inclusion graph of subgroups of $G$, denoted by $\mathcal I(G)$, is a graph having all the proper subgroups of $G$ as its vertices and two distinct vertices $H$ and $K$ in $\mathcal I(G)$ are adjacent…

We state and solve a discrete version of the classical Riemann-Hilbert problem. In particular, we associate a Riemann-Hilbert problem to every dessin d'enfants. We show how to compute the solution for a dessin that is a tree. This amounts…

For each finite group G, we define the Grothendieck-Teichm\"uller group of G, denoted GT(G), and explore its properties. The theory of dessins d'enfants shows that the inverse limit of GT(G) as G varies can be identified with a group…

Let $G$ be a group. The prime index graph of $G$, denoted by $\Pi(G)$, is the graph whose vertex set is the set of all subgroups of $G$ and two distinct comparable vertices $H$ and $K$ are adjacent if and only if the index of $H$ in $K$ or…

Riemann's Existence Theorem gives the following bijections: (1) Isomorphism classes of Belyi maps of degree $d$. (2) Equivalence classes of generating systems of degree $d$. (3) Isomorphism classes of dessins d'enfants with $d$ edges. In…

We investigate the representations and the structure of Hecke algebras associated to certain finite complex reflection groups. We first describe computational methods for the construction of irreducible representations of these algebras,…

We show that for certain arithmetic groups, geometrically finite subgroups are the intersection of finite index subgroups containing them. Examples are the Bianchi groups and the Seifert-Weber dodecahedral space. In particular, for…

Generalizing the dihedral picture for G(M,M,2), we construct Hecke algebras (and present a strategy for constructing Hecke categories) and asymptotic counterparts. We think of these as associated with the complex reflection group G(M,M,N).

We introduce a new invariant of bipartite chord diagrams and use it to construct the first examples of groups with Dehn function $n^2\log n$ and other small Dehn functions. Some of these groups have undecidable conjugacy problem.

Let $n$ and $k$ be integers with $n> k\geq1$ and $[n] = \{1, 2, ... , n\} $. The $bipartite \ Kneser \ graph$ $H(n, k)$ is the graph with the all $k$-element and all ($n-k$)-element subsets of $[n] $ as vertices, and there is an edge…

We introduce a family of models, which we name matrix models associated with children's drawings -- the so-called dessin d'enfant. Dessins d'enfant are graphs of a special kind drawn on a closed connected orientable surface (in the sky).…

Coverings of the Riemann sphere by itself, ramified over two points, are given by so-called Shabat polynomials. The correspondence between Grothendieck's dessins d'enfants and Belyi maps then implies a bijection between Shabat polynomials…