On the Pure Virtual Braid Group PV₃

Abstract

We investigate various properties of the pure virtual braid group PV₃. Out of its presentation, we get a free product decomposition of PV₃. As a consequence, we show that PV₃ is residually torsion free nilpotent, what implies that the set of the 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 PV₃ is aspherical. We determine also the cohomology ring and the associated graded Lie algebra of PV₃.

Key Words:

• Cohomology ring
• Lie algebra
• Presentation
• Pure virtual braid group
• Residual torsion free nilpotence

2010 Mathematics Subject Classification:

• Primary: 20F38
• Secondary: 20F36
• 20J06
• 57 M

ACKNOWLEDGMENTS

This work was started during the stay of the four authors in Oberwolfach in the framework of the programme RiP during May 18th–June 7th, 2008. The authors would like to thank the Mathematical Institute of Oberwolfach for its hospitality. The work was continued during the visit of the third author in the Institute for Mathematical Sciences of the National University of Singapore during December 4–19, 2008. He would like to express his gratitude to the IMS.