On the pure virtual braid group $PV_3$

TL;DR

This paper explores the algebraic and topological properties of the pure virtual braid group PV_3, revealing its structure, residual properties, asphericity, and cohomology, with implications for virtual braid invariants.

Contribution

It provides a canonical presentation, a free product decomposition, and computes the cohomology and Lie algebra of PV_3, advancing understanding of virtual braid groups.

Findings

PV_3 is residually torsion free nilpotent

PV_3's presentation is aspherical

Cohomology ring and graded Lie algebra of PV_3 are determined

Abstract

In this article, we investigate various properties of the pure virtual braid group PV_3. From its canonical presentation, we obtain a free product decomposition of PV_3. As a consequence, we show that PV_3 is residually torsion free nilpotent, which implies that the set of finite type invariants in the sense of Goussarov-Polyak-Viro is complete for virtual pure braids with three strands. Moreover we prove that the presentation of PV_3 is aspherical. Finally we determine the cohomology ring and the associated graded Lie algebra of PV_3.