Cluster braid groups of Coxeter-Dynkin diagrams

TL;DR

This paper introduces the exchange groupoid for finite Coxeter-Dynkin diagrams and proves its fundamental group is isomorphic to the associated braid group, linking algebraic and geometric structures.

Contribution

It extends the concept of cluster exchange groupoids to Coxeter-Dynkin diagrams and establishes a fundamental group isomorphism with the corresponding braid group.

Findings

The exchange groupoid for Coxeter-Dynkin diagrams is well-defined.

The fundamental group of this groupoid is isomorphic to the associated braid group.

Provides a new geometric perspective on Coxeter-Dynkin diagram braid groups.

Abstract

Cluster exchange groupoids are introduced by King-Qiu as an enhancement of cluster exchange graphs to study stability conditions and quadratic differentials. In this paper, we introduce the exchange groupoid for any finite Coxeter-Dynkin diagram $\Delta$ and show that the fundamental group of which is isomorphic to the corresponding braid group associated with $\Delta$.