Metacyclic groups as automorphism groups of compact Riemann surfaces

TL;DR

This paper establishes upper bounds on the size of automorphism groups of compact Riemann surfaces based on the structure of their Sylow subgroups, with specific bounds for cyclic and metacyclic groups, and demonstrates these bounds are sharp.

Contribution

It provides new bounds on automorphism group sizes for Riemann surfaces when the groups are cyclic or metacyclic, extending previous classifications.

Findings

Bound |G| ≤ 30(g-1) for groups with cyclic Sylow 2-subgroups.

Bound |G| ≤ 10(g-1) for groups with all Sylow subgroups cyclic, with two exceptions.

Bound |G| ≤ 12(g-1) for metacyclic groups, with one exception.

Abstract

Let $X$ be a compact Riemann surface of genus $g\geq 2$, and let $G$ be a subgroup of $Aut(X)$. We show that if the Sylow $2$-subgroups of $G$ are cyclic, then $|G|\leq 30(g-1)$. If all Sylow subgroups of $G$ are cyclic, then, with two exceptions, $|G|\leq 10(g-1)$. More generally, if $G$ is metacyclic, then, with one exception, $|G|\leq 12(g-1)$. Each of these bounds is attained for infinitely many values of $g$.