  In [V.O. Manturov, Non-reidemeister knot theory and its applications in
dynamical systems, geometry, and topology, arxiv:1501.05208] the first named
author gave the definition of $k$-free braid groups $G_n^k$. Here we establish
connections between free braid groups, classical braid groups and free groups:
we describe explicitly the homomorphism from (pure) braid group to $k$-free
braid groups for important cases $k=3,4$. On the other hand, we construct a
homomorphism from (a subgroup of) free braid groups to free groups. The
relations established would allow one to construct new invariants of braids and
to define new powerful and easily calculated complexities for classical braid
groups.
