Chiral Hypermaps of Small Genus

Antonio Breda D'Azevedo and Roman Nedela

Abstract: A hypermap $\cal H$ is a cellular embedding of a 3-valent graph $\cal G$ into a closed surface which cells are 3-coloured (adjacent cells have different colours). The vertices of $\cal G$ are called flags of $\cal H$ and let us denote by F the set of flags. An automorphism of the underlying graph which extends to a colour preserving self-homeomorphism of the surface is called an automorphism of the hypermap. If the surface is orientable the automorphisms of $\cal H$ split into two classes, orientation preserving and orientation reversing automorphisms. It is not difficult to observe that $|Aut({\cal H)}|\le |{\rm F}|$ while for the group of orientation preserving automorphisms we have $|Aut^+({\cal H})|\le |{\rm F}|/2$. A hypermap satisfying $|Aut^+({\cal H})|=|{\rm F}|/2=|Aut({\cal H})|$ will be called chiral. Hence chiral hypermaps have maximum number of orientation preserving symmetries but they are not ``mirror symmetric''.

The main goal of this paper is to classify all chiral hypermaps on surfaces of genus at most four. It follows that they consist of the infinite families of chiral toroidal hypermaps of types $(2,3,6)$, $(2,4,4)$, $(3,3,3)$, and their duals, and two exceptional chiral hypermaps (up to duality) of types $(3,3,7)$ and $(4,4,5)$. These exceptional chiral hypermaps are members of regular hypermaps with metacyclic oriented monodromy groups.

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