  We describe the construction of a minimal presentation for the group of
planar pure braids $\overline{P}_n$ on $n$ strands. The generators of this
presentation are dual to the generators of the cohomology ring of
$\overline{P}_n$ found by Y.Baryshnikov while the relations lie in the
commutator subgroup of $\overline{P}_n$. We also construct an explicit
(although non-minimal) presentation for the pure cactus group.
