Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 44 (2003), No. 1, 127-143.

Chiral Hypermaps of Small Genus

Antonio Breda D’Azevedo^{∗} Roman Nedela^{†}

Department of Mathematics, University Aveiro Aveiro,Portugal

School of Finance, Matej Bel University 975 49 Banská Bystrica, Slovakia

Abstract. A hypermap H is a cellular embedding of a 3-valent graph G into closed surface which cells are 3-coloured (adjacent cells have different colours). he vertices of G are called Ωags of H and let us denote by F the set of Ωags. n automorphism of the underlying graph which extends to a colour preserving elf-homeomorphism of the surface is called an automorphism of the hypermap. If he surface is orientable the automorphisms of H split into two classes, orientation reserving and orientation reversing automorphisms. It is not difficult to observe hat |Aut(H)| ≤ |F| while for the group of orientation preserving automorphisms e have |Aut^{+}(H)| ≤ |F|/2. A hypermap satisfying |Aut^{+}(H)| = |F|/2 = |Aut(H)| ⊒⟩↕↕ ⌊⌉ ⌋⊣↕↕⌉⌈ ⌋⟨⟩∇⊣↕↙ H⌉\⌋⌉ ⌋⟨⟩∇⊣↕ ⟨†√⌉∇⇕⊣√∫ ⟨⊣⊑⌉ ⇕⊣§⟩⇕⊓⇕ \⊓⇕⌊⌉∇ ≀{ ≀∇⟩⌉\⊔⊣⊔⟩≀\ ∇⌉∫⌉∇⊑⟩\} ∫†⇕⇕⌉⊔∇⟩⌉∫ ⌊⊓⊔ ⊔⟨⌉† ⊣∇⌉ \≀⊔ ∩⇕⟩∇∇≀∇ ∫†⇕⇕⌉⊔∇⟩⌋↑↙ ⟨⌉ ⇕⊣⟩\ }≀⊣↕ ≀{ ⊔⟨⟩∫ √⊣√⌉∇ ⟩∫ ⊔≀ ⌋↕⊣∫∫⟩{† ⊣↕↕ ⌋⟨⟩∇⊣↕ ⟨†√⌉∇⇕⊣√∫ ≀\ ∫⊓∇{⊣⌋⌉∫ ≀{ }⌉\⊓∫ ⊔ ⇕≀∫⊔ {≀⊓∇↙ I⊔ {≀↕↕≀⊒∫ ⊔⟨⊣⊔ ⊔⟨⌉† ⌋≀\∫⟩∫⊔ ≀{ ⊔⟨⌉ ⟩\⊙\⟩⊔⌉ {⊣⇕⟩↕⟩⌉∫ ≀{ ⌋⟨⟩∇⊣↕ ⊔≀∇≀⟩⌈⊣↕ †√⌉∇⇕⊣√∫ ≀{ ⊔†√⌉∫ ⇐∈, 3, 6), (2, 4, 4), (3, 3, 3), and their duals, and two exceptional hiral hypermaps (up to duality) of types (3, 3, 7) and (4, 4, 5). These exceptional hiral hypermaps are members of regular hypermaps with metacyclic oriented mondromy groups.

∗ Supported in part by UI&D “Matemática e aplicações”.

† Supported in part by Slovak Ministry for Education.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ⊗ \end{matrix} 2003 Heldermann Verlag

128 Antonio Breda D’Azevedo, Roman Nedela: Chiral Hypermaps of Small Genus 1. Introduction A map is a cellular decomposition of a closed surface. Maps on orientable surfaces can be described by means of two permutations R and L, with L being involutory, such that the group ⟨R, L⟩ acts transitively on the set D of darts of the map. The triple (D, R, L) determines the topological map up to isomorphism. If we relax the condition L^{2} = 1 by considering any couple R, L of permutations of D generating a transitive group G = ⟨R, L⟩ of permutations we end with an (algebraic) definition of an oriented hypermap H^{+} = (D, R, L). All the important notions such as genus, automorphism group, regularity and so on, extend naturally to hypermaps. The hypermap H is orientably regular if G acts regularly on D, and it is regular if it is orientably regular and the assignment R ↦ R^{−1} and L ↦ L^{−1} extends to a group automorphism. Orientably regular maps which are not regular will be called chiral. Orientably regular hypermaps are particularly nice objects since they make a bridge between geometry and algebra. In a sense, to study orientably regular hypermaps means to study two-generator groups with prescribed couples of generators. By the well-known Hurwitz bound the size of the group G = ⟨R, L⟩ of an orientably regular hypermap H = (D, R, L) is bounded by 84(g-1), where g > 1 is the genus. One of the central problems in the theory of maps and hypermaps is the problem of classification of all orientably regular hypermaps on a fixed underlying surface. Using the Euler formula it is not difficult to see that spherical orientably regular hypermaps consist of the five Platonic solids, and of two infinite families of types (1, n, n), (2, 2, n) and their duals. Orientably regular maps on torus were classified by Coxeter and Moser [11], and the generalisation to hypermaps was done by Corn and Singerman [10]. In [4] the classification problem for double torus is settled. As concerns surfaces of higher genera only partial results are known. For instance, Conder and Dobcsányi [9] with the help of a computer program gave a list of all orientably regular maps up to genus 15. In this paper we carry out the classification of chiral hypermaps with genus at most four. Chiral hypermaps form an interesting subset of the family of all orientably regular hyper- maps, see or instance [11, 17, 6, 5]. They have aximum number of orientation preserving automorphisms but they are not isomorphic with their mirror images. It is worth to mention that by examining the list of orientably regular maps up to genus 15 [9] one can see that surfaces of genera 2, 3, 4, 5, 6, 9 and 13 support no chiral maps. The main result of the paper implies that there is no chiral hypermap on surface of genus 2, while each of the surfaces of genus 3 and 4 supports (up to duality) exactly one chiral hypermap, both with metacyclic oriented monodromy group (see [5] for a description of an infinite family of chiral hypermaps with metacyclic oriented monodromy group). 2. Preliminaries A topological hypermap H is a cellular embedding of a connected trivalent graph G into a compact surface S, without boundary and not necessarily orientable, such that the cells are 3-coloured (say by black, grey and white colours) with adjacent cells having different colours.

Numbering the colours 0, 1 and 2, and labelling the edges of G with the missing adjacent cell number, we can define 3 fixed points free involutory permutations r_{i}, i = 0, 1, 2, on