Lifting Theorem for the Virtual Pure Braid Groups

In this article the authors prove theorem on Lifting for the set of virtual pure braid groups. This theorem says that if they know presentation of virtual pure braid group V P-4, then they can find presentation of V P(n )for arbitrary n > 4. Using this theorem they find the set of generators and defining relations for simplicial group T-* which was defined in [Bardakov, V. G. and Wu, J., On virtual cabling and structure of 4-strand virtual pure braid group, J. Knot Theory and Ram., 29(10), 2020, 1-32]. They find a decomposition of the Artin pure braid group P-n in semi-direct product of free groups in the cabled generators.