Regular dessins with a given automorphism group

Dessins d'enfants are combinatorial structures on compact Riemann surfaces defined over algebraic number fields, and regular dessins are the most symmetric of them. If G is a finite group, there are only finitely many regular dessins with automorphism group G. It is shown how to enumerate them, how to represent them all as quotients of a single regular dessin U(G), and how certain hypermap operations act on them. For example, if G is a cyclic group of order n then U(G) is a map on the Fermat curve of degree n and genus (n-1)(n-2)/2. On the other hand, if G=A_5 then U(G) has genus 274218830047232000000000000000001. For other non-abelian finite simple groups, the genus is much larger.Comment: 19 page

String C-group representations of alternating groups

We prove that for any integer n ≥ 12, and for every r in the interval [3, . . . , Floor((n−1)/2)], the group A_n has a string C-group representation of rank r, and hence that the only alternating group whose set of such ranks is not an interval is A_11.publishe