Garside elements, inertia and Galois action on braid groups

TL;DR

This paper investigates the Galois action on braid groups, emphasizing the cyclotomic nature of divisorial inertia, and extends the analysis to complex reflection groups and non-colored braid groups with stack-based classifying spaces.

Contribution

It provides a detailed analysis of Galois actions on profinite braid groups, including cases involving complex reflection groups and Deligne-Mumford stacks, highlighting the cyclotomic behavior of inertia.

Findings

Divisorial inertia acts cyclotomically on braid groups.

Extension of Galois action analysis to complex reflection groups.

Inclusion of non-colored braid groups with stack-based classifying spaces.

Abstract

An important piece of information in the theory of the arithmetic Galois action on the geometric fundamental groups of schemes is that divisorial inertia is acted on cyclotomically. We detail in this note the content of this fact in the case of the profinite braid groups arising from complex reflection groups, naturally viewing them as the geometric fundamental groups of the attending classifying spaces. We also include the case of the full (non colored) braid groups, whose completed classifying spaces are Deligne-Mumford stacks rather than schemes.