The pure braid groups and their relatives

TL;DR

This survey explores the algebraic and topological properties of pure braid groups and their relatives, including resonance varieties, ranks, and formality, highlighting their differences and relationships.

Contribution

It provides a comprehensive overview of the properties and interrelations of various braid-like groups, including new insights into their algebraic structures.

Findings

Resonance varieties and ranks vary among the groups.

Distinct properties help differentiate pure braid groups from their relatives.

Analysis of homomorphisms reveals structural relationships.

Abstract

In this mostly survey paper, we investigate the resonance varieties, the lower central series ranks, and the Chen ranks, as well as the residual and formality properties of several families of braid-like groups: the pure braid groups $P_n$, the welded pure braid groups $wP_n$, the virtual pure braid groups $vP_n$, as well as their `upper' variants, $wP_n^+$ and $vP_n^+$. We also discuss several natural homomorphisms between these groups, and various ways to distinguish among the pure braid groups and their relatives.