  Let $G$ be a locally compact, second countable, unimodular group that is
nondiscrete and noncompact. We explore the theory of invariant point processes
on $G$.
  We show that every free probability measure preserving (pmp) action of $G$
can be realized by an invariant point process.
  We analyze the cost of pmp actions of $G$ using this language. We show that
among free pmp actions, the cost is maximal on the Poisson processes. This
follows from showing that every free point process weakly factors onto any
Poisson process and that the cost is monotone for weak factors, up to some
restrictions. We apply this to show that $G\times \mathbb{Z}$ has fixed price
$1$, solving a problem of Carderi.
  We also show that when $G$ is a semisimple real Lie group, the rank gradient
of any Farber sequence of lattices in $G$ is dominated by the cost of the
Poisson process of $G$. This, in particular, implies that if the cost of the
Poisson process of $SL_{2}(\mathbb{C})$ vanishes, then the ratio of the
Heegaard genus and the rank of a hyperbolic $3$-manifold tends to infinity over
Farber chains.
