Several types of solvable groups as automorphism groups of compact Riemann surfaces

TL;DR

This paper establishes sharp upper bounds on the size of automorphism groups of compact Riemann surfaces for various classes of solvable groups, refining previous results and extending bounds to broader group types.

Contribution

It refines known bounds for automorphism groups by including groups of odd order and new solvable group types, and generalizes Zomorrodian's bound to a wider class of groups.

Findings

Bounds for groups of odd order are established.

Optimal bounds for groups of order p^m q^n are derived.

Zomorrodian's bound applies to any group with p as the smallest prime divisor.

Abstract

Let $X$ be a compact Riemann surface of genus $g\geq 2$. Let $Aut(X)$ be its group of automorphisms and $G\subseteq Aut(X)$ a subgroup. Sharp upper bounds for $|G|$ in terms of $g$ are known if $G$ belongs to certain classes of groups, e.g. solvable, supersolvable, nilpotent, metabelian, metacyclic, abelian, cyclic. We refine these results by finding similar bounds for groups of odd order that are of these types. We also add more types of solvable groups to that long list by establishing the optimal bounds for, among others, groups of order $p^m q^n$. Moreover, we show that Zomorrodian's bound for $p$-groups $G$ with $p\geq 5$, namely $|G|\leq \frac{2p}{p-3}(g-1)$, actually holds for any group $G$ for which $p\geq 5$ is the smallest prime divisor of $|G|$.