Regular Dessins Research Articles

Regular dessins with underlying graphs [formula omitted]: The metacyclic case

A dessin is a 2-cell embedding of a connected 2-colored bipartite graph into an orientable closed surface. A dessin is regular if its group of automorphisms, preserving both color and orientation, acts transitively on its edge set. Leveraging the theory of metacyclic 2-groups, we undertake the classification and enumeration of regular dessins with complete bipartite underlying graphs of the form K2e,2f, while imposing the additional criterion of metacyclic automorphism groups. Our findings extend and generalize the earlier work of Du et al. (2007) [5].

Uniquely regular dessins with nilpotent automorphism groups of odd prime power order

If a graph is cut along each edge and can be divided into many faces, the graph has been defined to be a map when each face is homomorphic to an open disk and a bipartite map called dessin. A dessin D is regular Aut (D) if action transfer on the edge set. In particular, given a finite group G, a regular dessin is uniquely regular dessin if there is only one isomorphism class of Aut (D) are isomorphic to G. In this paper, for a nilpotent automorphism group of odd prime power order and nilpotency class four, we employ group-theoretical methods to classify these uniquely regular dessins.

Regular dessins with moduli fields of the form Q(ζp, \sqrt[p]{q})

Gareth Jones asked during the 2014 SIGMAP conference for examples of regular dessins with nonabelian fields of moduli. In this paper, we first construct dessins whose moduli fields are nonabelian Galois extensions of the form Q(ζp, \sqrt[p]{q}), where p is an odd prime and ζp is a pth root of unity and q ∈ Q is not a pth power, and we then show that their regular closures have the same moduli fields. Finally, in the special case p = q = 3 we give another example of a regular dessin with moduli field Q(ζ3, ∛(3)) of degree 219 · 34 and genus 14155777.

Complete circular regular dessins of coprime orders

A complete dessin of order {m,n} is an orientable map with underlying graph being a complete bipartite graph Km,n, which is said to be regular if all edges are equivalent under the automorphism group, and circular if the boundary cycle of each face is a circuit. As one of a series papers towards a classification of complete circular regular dessins of order {m,n}, this paper presents such a classification for the case where m,n are coprime.

Dessins D’enfants and Some Holomorphic Structures on the Loch Ness Monster

Abstract The theory of dessins d’enfants on compact Riemann surfaces, which are bipartite maps on compact orientable surfaces, are combinatorial objects used to study branched covers between compact Riemann surfaces and the absolute Galois group of the field of rational numbers. In this paper, we show how this theory is naturally extended to non-compact orientable surfaces and, in particular, we observe that the Loch Ness monster (LNM; the surface of infinite genus with exactly one end) admits infinitely many regular dessins d’enfants (either chiral or reflexive). In addition, we study different holomorphic structures on the LNM, which come from homology covers of compact Riemann surfaces, and infinite hyperelliptic and infinite superelliptic curves.

Complete bipartite multi-graphs with a unique regular dessin

A regular dessin is an orientable edge-regular bipartite map. Jones et al.(J Combin Theory Ser B 98:241–248, 2008) showed that a complete bipartite graph $$\mathbf{K}_{n,n}$$ has a unique orientably (arc-)regular map if and only if $$\gcd (n,\phi (n))=1$$ . We extended this result in Fan and Li (J Graph Theory 87:581–586, 2018) by proving that a complete bipartite graph $$\mathbf{K}_{m,n}$$ underlies a unique regular dessin if and only if $$\gcd (m,\phi (n))=1$$ and $$\gcd (n,\phi (m))=1$$ . In this paper, it is shown that a complete bipartite multi-graph $$\mathbf{K}_{m,n}^{(\lambda )}$$ with $$\lambda >1$$ underlies a unique regular dessin if and only if $$\lambda =2$$ , $$\gcd (m_2,n_2)=1$$ , and $$\gcd (m,\phi (n_{2'}))\gcd (n,\phi (m_{2'}))=1$$ , where $$n_{2}$$ and $$n_{2'}$$ denote the 2-part and $$2'$$ -part of n, respectively. Furthermore, apart from two degenerated cases, each of such dessins can be uniquely decomposed into the direct product of two dessins such that one is symmetric and reflexible, and the other has only one face.

Complete regular dessins and skew-morphisms of cyclic groups

A dessin is a 2 -cell embedding of a connected 2 -coloured bipartite graph into an orientable closed surface. A dessin is regular if its group of orientation- and colour-preserving automorphisms acts regularly on the edges. In this paper we study regular dessins whose underlying graph is a complete bipartite graph K m , n , called ( m , n ) -complete regular dessins. The purpose is to establish a rather surprising correspondence between ( m , n ) -complete regular dessins and pairs of skew-morphisms of cyclic groups. A skew-morphism of a finite group A is a bijection φ : A → A that satisfies the identity φ ( x y ) = φ ( x ) φ π ( x ) ( y ) for some function π : A → ℤ and fixes the neutral element of A . We show that every ( m , n ) -complete regular dessin D determines a pair of reciprocal skew-morphisms of the cyclic groups ℤ n and ℤ m . Conversely, D can be reconstructed from such a reciprocal pair. As a consequence, we prove that complete regular dessins, exact bicyclic groups with a distinguished pair of generators, and pairs of reciprocal skew-morphisms of cyclic groups are all in a one-to-one correspondence. Finally, we apply the main result to determining all pairs of integers m and n for which there exists, up to interchange of colours, exactly one isomorphism class of ( m , n ) -complete regular dessins. We show that the latter occurs precisely when every group expressible as a product of cyclic groups of order m and n is abelian, which eventually comes down to the condition gcd ( m , ϕ ( n )) = gcd ( ϕ ( m ), n ) = 1 , where ϕ is Euler’s totient function.

Reflexible complete regular dessins and antibalanced skew morphisms of cyclic groups

A skew morphism of a finite group A is a bijection φ on A fixing the identity element of A and for which there exists an integer-valued function π on A such that φ(ab) = φ(a)φπ(a)(b), for all a, b ∈ A. In addition, if φ − 1(a) = φ(a − 1) − 1, for all a ∈ A, then φ is called antibalanced. In this paper we develop a general theory of antibalanced skew morphisms and establish a one-to-one correspondence between reciprocal pairs of antibalanced skew morphisms of the cyclic additive groups and isomorphism classes of reflexible regular dessins with complete bipartite underlying graphs. As an application, reflexible complete regular dessins are classified.

Counting the number of quasiplatonic topological actions of the cyclic group on surfaces

We derive a closed formula for the number of quasiplatonic topological actions of the cyclic group on surfaces of genus two or greater. The formula implies that the number of quasiplatonic cyclic group actions is roughly one-sixth the number of regular dessins d’enfants with a cyclic group of automorphisms.

Zapponi-orientable dessins d’enfants

Almost two decades ago, Zapponi introduced a notion of orientability of a clean dessin d’enfant, based on an orientation of the embedded bipartite graph. We extend this concept, which we call Z-orientability to distinguish it from the traditional topological definition, to the wider context of all dessins, and we use it to define a concept of twist orientability, which also takes account of the Z-orientability properties of those dessins obtained by permuting the roles of white and black vertices and face-centres. We observe that these properties are Galois-invariant, and we study the extent to which they are determined by the standard invariants such as the passport and the monodromy and automorphism groups. We find that in general they are independent of these invariants, but in the case of regular dessins they are determined by the monodromy group.

Regular dessins d'enfants with dicyclic group of automorphisms

Let Gn be the dicyclic group of order 4n. We observe that, up to isomorphisms, (i) for n≥2 even there is exactly one regular dessin d'enfant with automorphism group Gn, and (ii) for n≥3 odd there are exactly two of them. Each of them is produced on well known hyperelliptic Riemann surfaces. We obtain that the minimal genus over which Gn acts purely-non-free is σp(Gn)=n (this coincides with the strong symmetric genus of Gn when n is even). For each of the triangular conformal actions, every non-trivial subgroup of Gn has genus zero quotient, in particular, that the isotypical decomposition, induced by the action of Gn, of its jacobian variety has only one component. We also study conformal/anticonformal actions of Gn, on closed Riemann surfaces, with the property that Gn admits anticonformal elements. It is known that Gn always acts on a genus one Riemann surface with such a property. We observe that the next genus σhyp(Gn)≥2 over which Gn acts in that way is n+1 for n≥2 even, and 2n−2 for n≥3 odd. We also provide examples of pseudo-real Riemann surfaces admitting Gn as the full group of conformal/anticonformal automorphisms.

Complete regular dessins of odd prime power order

A dessin is a 2-cell embedding of a connected 2-coloured bipartite graph into an orientable closed surface. A dessin is regular if its group of colour- and orientation-preserving automorphisms acts regularly on the edges. In this paper we employ group-theoretic method to determine and enumerate the isomorphism classes of regular dessins with complete bipartite underlying graphs of odd prime power order.

Regular dessins uniquely determined by a nilpotent automorphism group

Abstract It is well known that the automorphism group of a regular dessin is a two-generator finite group, and the isomorphism classes of regular dessins with automorphism groups isomorphic to a given finite group G are in one-to-one correspondence with the orbits of the action of Aut ⁢ ( G ) {{\mathrm{Aut}}(G)} on the ordered generating pairs of G. If there is only one orbit, then up to isomorphism the regular dessin is uniquely determined by the group G and it is called uniquely regular. In this paper we investigate the classification of uniquely regular dessins with a nilpotent automorphism group. The problem is reduced to the classification of finite maximally automorphic p-groups G, i.e., the order of the automorphism group of G attains Hall’s upper bound. Maximally automorphic p-groups of nilpotency class three are classified.

Regular dessins d'enfants with field of moduli Q(p√2)

Herradon has recently provided an example of a regular dessin d’enfant whose field of moduli is the non-abelian extension Q ( 3 √2) answering in this way a question due to Conder, Jones, Streit and Wolfart. In this paper we observe that Herradon’s example belongs naturally to an infinite series of such kind of examples; for each prime integer p ≥ 3 we construct a regular dessin d’enfant whose field of moduli is the non-abelian extension Q ( p √2) ; for p = 3 it coincides with Herradon’s example.

An explicit quasiplatonic curve with non-abelian moduli field

We give an example of a regular dessin d'enfant whose field of moduli is not an abelian extension of the rational numbers, namely it is the field generated by a cubic root of 2. This answers a previous question. We also prove that the underlying curve has non-abelian field of moduli itself, giving an explicit example of a quasiplatonic curve with non-abelian field of moduli. In the last section, we note that two examples in previous literature can be used to find other examples of regular dessins d'enfants with non-abelian field of moduli.

The absolute Galois group acts faithfully on regular dessins and on Beauville surfaces

In this article we study the action of the absolute Galois group Gal(Q/Q) on dessins d’enfants and Beauville surfaces. A foundational result in Grothendieck’s theory of dessins d’enfants is the fact that the absolute Galois group Gal(Q/Q) acts faithfully on the set of all dessins. However the question of whether this holds true when the action is restricted to the set of the, more accessible, regular dessins seems to be still an open question. In the first part of this paper we give an affirmative answer to it. In fact we prove the strongest result that the action is faithful on the set of quasiplatonic (or triangle) curves of any given hyperbolic type. Beauville surfaces are an important kind of algebraic surfaces introduced by Catanese. They are rigid surfaces of general type closely related to dessins d’enfants. Here we prove that for any σ ∈ Gal(Q/Q) different from the identity and the complex conjugation there is a Beauville surface S such that S and its Galois conjugate S have non-isomorphic fundamental groups. This in turn easily implies that the action of Gal(Q/Q) on the set of Beauville surfaces is faithful. These results were conjectured by Bauer, Catanese and Grunewald, and immediately imply that Gal(Q/Q) acts faithfully on the connected components of the moduli space of surfaces of general type, a result due to the above mentioned authors.

An elementary approach to dessins d’enfants and the Grothendieck–Teichmüller group

We give an account of the theory of dessins d'enfants which is both elementary and self-contained. We describe the equivalence of many categories (graphs embedded nicely on surfaces, finite sets with certain permutations, certain field extensions, and some classes of algebraic curves), some of which are naturally endowed with an action of the absolute Galois group of the rational field. We prove that the action is faithful. Eventually we prove that Gal (\overline {\mathbb Q}/\mathbb Q) embeds into the Grothendieck-Teichmüller group \widehat {\mathcal {GT}_0} introduced by Drinfeld. There are explicit approximations of \widehat {\mathcal {GT}_0} by finite groups, and we hope to encourage computations in this area. Our treatment includes a result which has not appeared in the literature yet: the action of Gal (\overline {\mathbb Q}/\mathbb Q) on the subset of regular dessins – that is, those exhibiting maximal symmetry – is also faithful.

Nilpotent groups of class two which underly a unique regular dessin

A dessin is an embedding of connected bipartite graph into an oriented closed surface. A dessin is regular if its group of colour- and orientation-preserving automorphisms acts transitively on the edges. In the present paper regular dessins with a nilpotent automorphism group are investigated, and attention are paid on those with the highest level of external symmetry. Depending on the algebraic theory of dessins and using group-theoretical methods, we present a classification of nilpotent groups of class two which underly a unique regular dessin.

Maps of Archimedean class and operations on dessins

In the present paper we introduce a family of functors (called operations) of the category of hypermaps (dessins) preserving the underlying Riemann surface. The considered family of functors include as particular instances the operations considered by Magot and Zvonkin (2000), Singerman and Syddall (2003), and Girondo (2003). We identify a set of 10 operations in the above infinite family which produce vertex-transitive dessins out of regular ones. This set is complete in the following sense: if a vertex-transitive map arises from a regular dessin H applying an operation, then it can be obtained from a regular dessin on the same surface (possibly different from H) applying one of the 10 operations. The statement includes the classical case when the underlying surface is the sphere.

Galois actions on regular dessins of small genera

Dessins d'enfants can be regarded as bipartite graphs embedded in compact orientable surfaces. According to Grothendieck and others, a dessin uniquely determines a complex structure on the surface, and even an algebraic structure (as a projective algebraic curve defined over a number field). The general problem of how to determine all properties of the curve from the combinatorics of the dessin is far from being solved. For regular dessins, which are those having an edge-transitive automorphism group, the situation is easier: currently available methods in combinatorial and computational group theory allow the determination of the fields of definition for all curves with regular dessins of genus 2 to 18.