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 ergodic theory of invariant point processes on G. Our first result shows that every free probability measure preserving (pmp) action of G can be realized by an invariant point process. We then 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 × 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. The same holds for the symmetric space X of G. This, in particular, implies that if the cost of the Poisson process of the hyperbolic 3-space ℍ3 vanishes, then the ratio of the Heegaard genus and the rank of a hyperbolic 3-manifold tends to infinity over arbitrary expander Farber sequences, in particular, the ratio can get arbitrarily large. On the other hand, if the cost of the Poisson process on ℍ3 does not vanish, it solves the cost versus L2 Betti problem of Gaboriau for countable equivalence relations.