# Galois action on homology of generalized Fermat Curves

## TL;DR

This paper studies the Galois actions on the homology of generalized Fermat curves, revealing their structure and connections to Galois representations and the Burau representation.

## Contribution

It computes the fundamental group of these curves and analyzes the Galois and Galois cover actions on their homology, unifying various aspects of their symmetry.

## Key findings

- Computed the fundamental group of generalized Fermat curves.
- Analyzed Galois and absolute Galois group actions on homology.
- Explored connections to the pro-ℓ Burau representation.

## Abstract

The fundamental group of Fermat and generalized Fermat curves is computed. These curves are Galois ramified covers of the projective line with abelian Galois groups $H$. We provide a unified study of the action of both cover Galois group $H$ and the absolute Galois group $\mathrm{Gal}(\bar{\Q}/\Q)$ on the pro-$\ell$ homology of the curves in study. Also the relation to the pro-$\ell$ Burau representation is investigated.

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Source: https://tomesphere.com/paper/1905.04298