On the orders of largest groups of automorphisms of compact Riemann surfaces

Let mu(g) be the largest possible order of a group of automorphisms of a compact Riemann surface of genus g >= 2. The celebrated results of Accola-Maclachlan and Hurwitz-Macbeath, taken together say that mu(g) ranges between 8(g+1) and 84(g-1) and these two bounds are attained for infinitely many g. Here we prove that given a prime q> 167congruent to -1modulo 3, not congruent +1modulo 5and a prime p >= 84q, mu(g) = -8(q+ 4)/ q)(g- 1) for g= qp(m)+ 1for infinitely many integers m. Furthermore we also prove that for any prime qfor which p = q+ 2is a prime, mu(g) =(12(q+ 2)/q)(g- 1) for g= qp(m)+ 1for infinitely many m. These mean that there exist an infinite rational sequence (r(n)) convergent to 8and, assuming the truth of the Twin Prime Conjecture, an infinite rational sequence (s(n)) convergent to 12together with infinite sets R-n and S n of integers for which mu(g) = {r(n)(g - 1) if g is an element of R-n s(n) (g - 1) if g is an element of S-n and 8, 12 are the unique numbers for which this can happen. A consequence of these results is the full understanding of the asymptotic's of mu. Namely if instead of mu = mu(g) we consider its normalization (mu) over tilde (g) = mu(g)/g then {8} subset of (M-d)(d) subset of {8, 12}, where M stands for the set of values of (mu) over tilde and the operator d for the set of accumulation points and, assuming the truth of the Twin Prime Conjecture, 12 is an element of (M-d)(d). (c) 2021 Elsevier B.V. All rights reserved.