How often is \(84(g-1)\) achieved? (Q5957363)

It is a well-known theorem of Hurwitz that a Riemann surface of genus \(g\geq 2\) has at most \(84(g-1)\) automorphisms. Macbeath showed that this bound is attained for infinitely many values of \(g\) and Accola proved that it fails to be attained infinitely often. The author looks at the question of how often it is achieved. It is proved that good values of \(g\) are about as common as perfect cubes. More precisely, let \(H\) denote the set of integers \(g\geq 2\) such that there exists at least one compact Riemann surface of genus \(g\) with automorphism group of order \(84(g-1)\). Then the series \(\sum_{g\in H}g^{-s}\) converges absolutely for \(\text{Re}(s)>1/3\) and has a singularity at \(s=1/3\). For any finitely generated group \(\Gamma\), the asymptotics of the set of orders of finite quotient groups of \(\Gamma\) are determined by the minimum dimension of a complex linear group containing an infinite quotient of \(\Gamma\). A proof and an application to the asymptotic behavior of the set of integers g for which the Hurwitz bound is sharp are given.