A four-term exact sequence of fundamental groups of orbit configuration spaces

TL;DR

This paper establishes a four-term exact sequence relating the fundamental groups of orbit configuration spaces under certain group actions on aspherical 2-manifolds, extending known orbifold pure braid group sequences.

Contribution

It introduces a new four-term exact sequence for fundamental groups of orbit configuration spaces, linking them to orbifold pure braid groups.

Findings

Fundamental groups fit into a four-term exact sequence.

The sequence relates orbit configuration spaces to orbifold pure braid groups.

New consequences for orbifold pure braid group sequences are derived.

Abstract

We deduce that the fundamental groups of the orbit configuration spaces of an effective and properly discontinuous action of a discrete group on a connected aspherical 2-manifold, with isolated fixed points, fit into a four-term exact sequence. This comes as a consequence of the four-term exact sequence of orbifold pure braid groups ([18], [11] and [19]). The proof relates these two exact sequences and also draws a new consequence (Corollary 2.3) on the later one.