On the exponent of the automorphism group of a compact Riemann surface

TL;DR

This paper establishes an upper bound of 42(g-1) for the exponent of the automorphism group of a compact Riemann surface of genus g, explicitly characterizes cases reaching this bound, and explores related subgroup properties.

Contribution

It provides a new universal bound on the automorphism group's exponent and explicitly identifies all cases where this bound is attained.

Findings

Bound of 42(g-1) for automorphism group exponent

Explicit classification of genus g where bound is reached

Discussion of subgroup properties like solvability

Abstract

Let $X$ be a compact Riemann surface of genus $g\geq 2$, and let $Aut(X)$ be its group of automorphims. We show that the exponent of $Aut(X)$ is bounded by $42(g-1)$. We also determine explicitly the infinitely many values of $g$ for which this bound is reached and the corresponding groups. Finally we discuss related questions for subgroups $G$ of $Aut(X)$ that are subject to additional conditions, for example being solvable.