Point processes, cost, and the growth of rank in locally compact groups

Abstract: 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.