Harvey-Wiman hypermaps

AbstractWe show that on a hypermap (α,σ) of genus g ⩾ 2, an automorphism ψ is either order o(ψ) = p(1 + 2g/(p − 1)) if (p, 1 + 2g/(p − 1)) = 1 or o(ψ) ⩽ 2pg/(p − 1), where p is the smallest divisor of the order of Aut(α,σ). We also give bounds on |Aut(α,σ)|, namely, |Aut(α,σ) ⩽ 2p(g − 1)/(p − 3) if p ⩾ 5, |Aut(α,σ)| ⩽ 15(g − 1) if p = 3; thus, only when p = 2 is the Hurwitz bound |Aut(α,σ)| ⩽ 84(g − 1) effective. We define p-Harvey hypermaps as hypermaps admitting an automorphism of order p(1 + 2g/(p − 1)) (type I) or 2pg/(p − 1) (type II) and characterise them as p-elliptic hypermaps

Symmetries of quasiplatonic Riemann surfaces

We state and prove a corrected version of a theorem of Singerman, which relates the existence of symmetries (anticonformal involutions) of a quasiplatonic Riemann surface $\mathcal S$ (one uniformised by a normal subgroup $N$ of finite index in a cocompact triangle group $\Delta$) to the properties of the group $G=\Delta/N$. We give examples to illustrate the revised necessary and sufficient conditions for the existence of symmetries, and we relate them to properties of the associated dessins d'enfants, or hypermaps

Fixed points of induced automorphisms of hypermaps

AbstractWe prove a formula that gives a relationship between the number of fixed points of an automorphism of a quotient hypermap and that of its liftings

Maps on surfaces and Galois groups

A brief survey of some of the connections between maps on surfaces, permutations, Riemann surfaces, algebraic curves and Galois groups is given

Quantum contextual finite geometries from dessins d'enfants

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 point-line configurations. These include simplices, cross-polytopes, several notable projective configurations, a number of multipartite graphs and some 'exotic' geometries. Among them, remarkably, we find not only those underlying Mermin's magic square and magic pentagram, but also those related to the geometry of two- and three-qubit Pauli groups. Of particular interest is the occurrence of all the three types of slim generalized quadrangles, namely GQ(2,1), GQ(2,2) and GQ(2,4), and a couple of closely related graphs, namely the Schl\"{a}fli and Clebsch ones. These findings seem to indicate that {\it dessins d'enfants} may provide us with a new powerful tool for gaining deeper insight into the nature of finite-dimensional Hilbert spaces and their associated groups, with a special emphasis on contextuality.Comment: 18 page

Universal Tessellations

Sin resume

Hypermaps: constructions and operations

It is conjectured that given positive integers l, m, n with l-1 + m-1 + n-1 < 1and an integer g ≥ 0, the triangle group Δ = Δ (l, m, n) = ⟨X,Y,Z|X l = Y m =Z n = X Y Z = 1⟩ contains infinitely many subgroups of finite index and of genusg. This conjecture can be rewritten in another form: given positive integers l,m, n with l¡1 +m¡1 +n¡1 < 1 and an integer g ≥ 0, there are infinitely manynonisomorphic compact orientable hypermaps of type (l, m, n) and genus g.We prove that the conjecture is true, when two of the parameters l, m, n areequal, by showing how to construct those hypermaps, and we extend the resultto nonorientable hypermaps.A classification of all operations of finite order in oriented hypermaps isgiven, and a detailed study of one of these operations (the duality operation)is developed. Adapting the notion of chirality group, the duality group ofH can be defined as the minimal subgroup D(H) ≤¦ M on (H) such thatH = D (H) is a self-dual hypermap. We prove that for any positive integer d,we can find a hypermap of that duality index (the order of D (H) ), even whensome restrictions apply, and also that, for any positive integer k, we can find anon self-dual hypermap such that |Mon (H) | = d = k. We call this k the dualitycoindex of the hypermap. Links between duality index, type and genus of aorientably regular hypermap are explored.Finally, we generalize the duality operation for nonorientable regular hypermaps and we verify if the results about duality index, obtained for orientably regular hypermaps, are still valid