  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.
