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Register a new userMonodromy Groups of Dessins dEnfant on Rational Triangular Billiards Surfaces
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A dessin denfant, or dessin, is a bicolored graph embedded into a Riemann surface, and the monodromy group is an algebraic invariant of the dessin generated by rotations of edges about black and white vertices. A rational billiards surface is a two dimensional surface that allows one to view the path of a billiards ball as a continuous path. In this paper, we classify the monodromy groups of dessins associated to rational triangular billiards surfaces.
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We present a number of examples to illustrate the use of small quotient dessins as substitutes for their often much larger and more complicated Galois (minimal regular) covers. In doing so we employ several useful group-theoretic techniques, such as the Frobenius character formula for counting triples in a finite group, pointing out some common traps and misconceptions associated with them. Although our examples are all chosen from Hurwitz curves and groups, they are relevant to dessins of any type.
We study general properties of the dessins denfants associated with the Hecke congruence subgroups $Gamma_0(N)$ of the modular group $mathrm{PSL}_2(mathbb{R})$. The definition of the $Gamma_0(N)$ as the stabilisers of couples of projective lattices in a two-dimensional vector space gives an interpretation of the quotient set $Gamma_0(N)backslashmathrm{PSL}_2(mathbb{R})$ as the projective lattices $N$-hyperdistant from a reference one, and hence as the projective line over the ring $mathbb{Z}/Nmathbb{Z}$. The natural action of $mathrm{PSL}_2(mathbb{R})$ on the lattices defines a dessin denfant structure, allowing for a combinatorial approach to features of the classical modular curves, such as the torsion points and the cusps. We tabulate the dessins denfants associated with the $15$ Hecke congruence subgroups of genus zero, which arise in Moonshine for the Monster sporadic group.
A short review of the Operator/Feynman diagram/dessin denfants correspondence in the rank 3 tensor model is presented, and the cut & join operation is given in the language of dessin denfants as a straightforward development. We classify operators of the rank 3 tensor model up to level 5 with dessin denfants.
We investigate the average number of solutions of certain quadratic congruences. As an application, we establish Manins conjecture for a cubic surface whose singularity type is A_5+A_1.
There have been several constructions of family of varieties with exceptional monodromy group. In most cases, these constructions give Hodge structures with high weight(Hodge numbers spread out). N. Katz was the first to obtain Hodge structures with low weight(Hodge numbers equal to (2,3,2)) and geometric monodromy group G2. In this article I will give an alternative description of Katzs construction and give an extension of his result.
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