  Let X be a Poisson point process of intensity lambda on the real line. A
thickening of it is a (deterministic) measurable function f such that the union
of X and f(X) is a Poisson point process of intensity lambda' where
lambda'>lambda. An equivariant thickening is a thickening which commutes with
all shifts of the line. We show that a thickening exists but an equivariant
thickening does not. We prove similar results for thickenings which commute
only with integer shifts and in the discrete and multi-dimensional settings.
This answers 3 questions of Holroyd, Lyons and Soo.
  We briefly consider also a much more general setup in which we ask for the
existence of a deterministic coupling satisfying a relation between two
probability measures. We present a conjectured sufficient condition for the
existence of such couplings.
