Groups of automorphisms of Riemann surfaces and maps of genus p + 1 where p is prime

Abstract

We classify compact Riemann surfaces of genus g, where g − 1 is a prime p, which have a group of automorphisms of order ρ(g − 1) for some integer ρ ≥ 1, and determine isogeny decompositions of the corresponding Jacobian varieties. This extends results of Belolipetzky and the second author for ρ > 6, and of the first and third authors for ρ = 3, 4, 5 and 6. As a corollary we classify the orientably regular hypermaps (including maps) of genus p + 1, together with the non-orientable regular hypermaps of characteristic −p, with automorphism group of order divisible by the prime p; this extends results of Conder, Širáň and Tucker for maps.

A computer-free classification of orientably-regular maps on surfaces of genus p+1 for prime p

“…Next, |G| = 2µp must be a multiple of the order of some non-abelian simple group H, with |H| ≤ |G| ≤ 84 • 83 = 6972. There are exactly thirteen such simple groups, namely A n for n ∈ {5, 6, 7}, PSL(2, q) for q ∈ {7, 8, 11, 13,16,17,19,23], PSL (3,3) and PSU(3, 3), with orders 60, 360, 2520, 168, 504, 660, 1092, 4080, 2448, 3420, 6072, 5616 and 6048 respectively. (This may be checked at numerous online references; for an article reference pre-dating the Classification of Finite Simple Groups see [16].…”

“…By our summary this amounts to supplying computer-free arguments for classification of orientably-regular maps M with |Aut + (M )| divisible by p for primes p in the range 17 ≤ p ≤ 83 as considered seperately in [12], but our approach will be more general and valid for all primes p ≥ 5. A task like this has recently been accomplished also by Izquierdo, Jones and Reyes-Carocca [17], in a computer-free classification of orientablyregular hypermaps of genus p + 1 with the orientation-preserving automorphism group having order divisible by p, for primes p ≥ 5 (but with no counterpart for the case where the group order is relatively prime to p). Their arguments, however, are based on the theory of Riemann surfaces and go beyond elementary group theory, whereas our approach emphasises easy accessibility.…”