Pure virtual braids, resonance, and formality

TL;DR

This paper studies algebraic invariants of pure virtual braid groups, establishing their 1-formality and exploring connections between their invariants and associated graded Lie algebras.

Contribution

It provides a complete answer to the 1-formality of pure virtual braid groups and explores the relationship between Alexander invariants and graded Lie algebras.

Findings

Pure virtual braid groups are 1-formal.

Resonance and Chen ranks are computed for these groups.

Connections between Alexander invariants and graded Lie algebras are discussed.

Abstract

We investigate the resonance varieties, lower central series ranks, and Chen ranks of the pure virtual braid groups and their upper-triangular subgroups. As an application, we give a complete answer to the 1-formality question for this class of groups. In the process, we explore various connections between the Alexander-type invariants of a finitely generated group and several of the graded Lie algebras associated to it, and discuss possible extensions of the resonance-Chen ranks formula in this context.