Pseudo-Hurwitz maps on alternating and symmetric groups

Hurwitz groups are the non-trivial finite quotients of the (2,3,7)-triangle group < A, Z| A(2) = Z(3) = (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) = -42 chi for given genus g >= 2(or given Euler characteristic chi - 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 gof the underlyinghyperbolic 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| A(2) = Z(3) =[A, Z](4) = 1 >, and as such, are the automorphism groups of regular bi-oriented maps of largest possible order 48(g-1) = -24 chi for given chi = 2 - 2g <= -2. In this paper we prove that the alternating group A(n) and the symmetric group S-n are pseudo-Hurwitz groups for all but a small number positive integers n, and in particular, for all n > 23. (c) 2023 Elsevier Inc. All rights reserved.