Riemann surfaces and restrictively-marked hypermaps

Abstract

If S is a compact Riemann surface of genus g > 1 then S has at most 84(g − 1) (orientation preserving) automorphisms (Hurwitz). On the other hand, if G is a group of automorphisms of S and |G| > 24(g − 1) then G is the automorphism group of a regular oriented map (of genus g) and if |G| > 12(g − 1) then G is the automorphism group of a regular oriented hypermap of genus g (Singerman). We generalise these results and prove that if |G| > g − 1 then G is the automorphism group of a regular restrictedly-marked hypermap of genus g. As a special case we also show that a marked finite transitive permutation group (Singerman) is a restrictedly-marked hypermap with the same genus.