Configuration Lie groupoids and orbifold braid groups

TL;DR

This paper introduces new definitions of configuration Lie groupoids, proves fibration theorems, and explores their fundamental groups, revealing structural properties of orbifold braid groups and affine Artin groups, with implications for the Farrell-Jones conjecture.

Contribution

It defines configuration Lie groupoids, proves fibration theorems, and establishes the poly-virtually free structure of orbifold braid groups and certain Artin groups, advancing understanding of their algebraic properties.

Findings

Pure orbifold braid groups are poly-virtually free.

Explicit generators for pure orbifold braid groups are provided.

The Farrell-Jones conjecture holds for the studied orbifold braid groups.

Abstract

We propose two definitions of configuration Lie groupoids and in both the cases we prove a Fadell-Neuwirth type fibration theorem for a class of Lie groupoids. We show that this is the best possible extension, in the sense that, for the class of Lie groupoids corresponding to global quotient orbifolds with nonempty singular set, the fibration theorems do not hold. Secondly, we prove a short exact sequence of fundamental groups (called {\it pure orbifold braid groups}) of one of the configuration Lie groupoids of the Lie groupoid corresponding to the punctured complex plane with cone points. This shows the possibility of a quasifibration type Fadell-Neuwirth theorem for Lie groupoids. As consequences, first we see that the pure orbifold braid groups have poly-virtually free structure, which generalizes the classical braid group case. We also provide an explicit set of generators of the pure orbifold braid groups. Secondly, we prove that a class of affine and finite complex Artin groups are virtually poly-free, which partially answers the question if all Artin groups are virtually poly-free ([[4], Question 2]). Finally, combining this poly-virtually free structure and a recent result ([5]), we deduce the Farrell-Jones isomorphism conjecture for the above class of orbifold braid groups. This also implies the conjecture for the case of the affine Artin group of type $\tilde D_n$, which was left open in [[24], Problem].