Characters of the Grothendieck-Teichmueller group through rigidity of the Burau representation

TL;DR

This paper demonstrates how characters of absolute Galois groups can be recovered via their action on profinite braid groups using the Burau representation, extending to Grothendieck-Teichmueller groups.

Contribution

It introduces a rigidity-based method to recover Galois characters through classical braid group representations, notably the Burau representation, and extends these to Grothendieck-Teichmueller groups.

Findings

Characters of Galois groups can be recovered from braid group automorphisms.

The Burau representation plays a key role in this recovery process.

Extension of Galois characters to Grothendieck-Teichmueller groups is achieved.

Abstract

We present examples of characters of absolute Galois groups of number fields that can be recovered through their action by automorphisms on the profinite completion of the braid groups, using a ``rigidity'' approach. The way we use to recover them is through classical representations of the braid groups, and in particular through the Burau representation. This enables one to extend these characters to Grothendieck-Teichmueller groups.