Pseudo-Hurwitz maps on alternating and symmetric groups

Abstract

Hurwitz groups are the non-trivial finite quotients of the (2,3,7)-triangle group 〈A,Z|A2=Z3=(AZ)7=1〉, with the name ‘Hurwitz’ coming from the fact that they are the conformal automorphism groups of compact Riemann surfaces of largest possible order 84(g−1)=−42χ for given genus g⩾2 (or given Euler characteristic χ−2−2g). As such, they are also the orientation-preserving automorphism groups of orientably-regular maps of largest possible order 84(g−1) for given genus g of the underlying hyperbolic surface, namely the ‘Hurwitz maps’, of type {3,7} or {7,3}.Pseudo-Hurwitz groups are the finite smooth quotients of the pseudo triangle group 〈A,Z|A2=Z3=[A,Z]4=1〉, and as such, are the automorphism groups of regular bi-oriented maps of largest possible order 48(g−1)=−24χ for given χ=2−2g≤−2. In this paper we prove that the alternating group An and the symmetric group Sn are pseudo-Hurwitz groups for all but a small number positive integers n, and in particular, for all n>23.