Group Actions on cyclic covers of the projective line

TL;DR

This paper employs combinatorial group theory to analyze the actions of automorphism groups, mapping class groups, and the Galois group on cyclic covers of the projective line, revealing structural insights into these symmetries.

Contribution

It introduces a novel approach using combinatorial group theory to study group actions on cyclic covers of the projective line, connecting geometric and algebraic perspectives.

Findings

Characterization of automorphism group actions on cyclic covers

Insights into Galois group actions on these covers

New connections between group actions and surface topology

Abstract

We use tools from combinatorial group theory in order to study actions of three types on groups acting on a curve, namely the automorphism group of a compact Riemann surface, the mapping class group acting on a surface (which now is allowed to have some points removed) and the absolute Galois group $\mathrm{Gal}(\bar{\Q}/\Q)$ in the case of cyclic covers of the projective line.