Regular dessins d'enfants with dicyclic group of automorphisms

摘要

Let G(n) 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 G(n), 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 G(n), acts purely-non-free is sigma(p)(G(n)) = n (this coincides with the strong symmetric genus of G(n) when n is even). For each of the triangular conformal actions, every non-trivial subgroup of G(n) has genus zero quotient, in particular, that the isotypical decomposition, induced by the action of G(n), of its jacobian variety has only one component. We also study conformal/anticonformal actions of G(n), on closed Riemann surfaces, with the property that G(n) admits anticonformal elements. It is known that G(n) always acts on a genus one Riemann surface with such a property. We observe that the next genus sigma(hyp)(G(n)) >= 2 over which G(n) 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 G(n) as the full group of conformal/anticonformal automorphisms. (C) 2019 Elsevier B.V. All rights reserved.