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 $P_n$, the welded pure braid groups $w{P}_n$, the virtual pure braid groups $v{P}_n$, as well as their `upper' variants, $w{P}_n^{+}$ and $v{P}_n^{+}$. We also discuss several natural homomorphisms between these groups, and various ways to distinguish among the pure braid groups and their relatives.

arXiv comment24 pages; expanded version, to appear in the proceedings of the intensive research period on Perspectives in Lie theory (Pisa, 2015), Springer INdAM series

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Cited by in corpus (6)

• Formality properties of finitely generated groups and Lie algebras
• Pure virtual braids, resonance, and formality
• Cup products, lower central series, and holonomy Lie algebras
• Chen ranks and resonance varieties of the upper McCool groups
• When the lower central series stops: a comprehensive study for braid groups and their relatives
• Taylor expansions of groups and filtered-formality