The pure braid groups and their relatives

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 Pn, the welded pure braid groups wPn, the virtual pure braid groups vPn, as well as their ‘upper’ variants, wPn+ and vPn+. We also discuss several natural homomorphisms between these groups, and various ways to distinguish among the pure braid groups and their relatives.