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Regular maps can be considered as special types of regular dessins. They include some of the oldest geometric objects known to mankind, in the form of the Platonic solids. These, together with the dihedra and their duals, the hosohedra (meaning ‘with many faces’), are the regular maps on the sphere. In this chapter we study their generalisations to maps on compact Riemann surfaces of arbitrary genus. We show how to classify regular maps in terms of their genus, paying particular attention to the cases of genus 0—as above—and of genus 1, where there are infinitely many regular maps, associated with ideals in the rings of Gaussian and Eisenstein integers. For each genus g ≥ 2 the Hurwitz bound implies that there are only finitely many regular maps, and we briefly consider the classification for g = 2. We also outline how one can classify regular maps in terms of their automorphism group, and we consider which groups can arise in this context.
KeywordsAutomorphism group Hurwitz group Quasiplatonic surface Regular map
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