A Riemann-Hurwitz Formula for Hypermaps Congruences

Algebraic Hypermap Morphisms

“…Therefore, the above sum equals the total branch number of . Cacciari [1] proved the Riemann-Hurwitz formula for a quotient hypermap with respect to a congruence, that is, an equivalence relation on a hypermap H which is compatible with the action of its group G(H). If ∼ is a congruence on H, then one can construct in an obvious way a quotient hypermap H ∼ and a morphism from H to H ∼ ; and vice versa, if : H → H is any morphism, then it determines a congruence ∼ on H such that H ∼ is isomorphic to H .…”

“…It turns out that algebraic hypermaps enjoy several properties of Riemann surfaces. For instance, each algebraic hypermap has its genus, and the analogue of Riemann-Hurwitz formula for Riemann surfaces was proved to hold for a quotient hypermap with respect to a group of automorphisms by Machì [10], and, more generally, with respect to a congruence by Cacciari [1]. Furthermore, topological hypermaps are essentially what Grothendieck called dessins d'enfants, and as such they are strictly related to Belyȋ functions, so that there exists a faithful action of the absolute Galois group of the rationals on them (see [7][8][9]11]).…”