Riemann surfaces and restrictively-marked hypermaps

english

2010

volume 3, issue 1

Univerza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije
(Obvezni izvod spletne publikacije)

If ?$\mathcal{S}$? is a compact Riemann surface of genus ?$g > 1$?, then ?$\mathcal{S}$? has at most ?$84(g - 1)$? (orientation preserving) automorphisms (Hurwitz). On the other hand, if ?$G$? is a group of automorphisms of ?$\mathcal{S}$? and ?$\vert G \vert > 24(g - 1)$? then ?$G$? is the automorphism group of a regular oriented map (of genus ?$g$?) and if ?$\vert G \vert > 12(g - 1)$? then ?$G$? is the automorphism group of a regular oriented hypermap of genus ?$g$? (Singerman). We generalise these results and prove that if ?$\vert G \vert > g - 1$? 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.

16.12.2021