No 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 × 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 (C) vanishes, then the ratio of the Heegaard genus and the rank of a hyperbolic 3-manifold tends to infinity over Farber chains.

We define a probability measure preserving and r-discrete groupoid that is associated to every invariant point process on a locally compact and second countable group. This groupoid governs certain factor processes of the point process, in particular the existence of Cayley factor graphs. With this method we are able to show that point processes on amenable groups admit all (and only admit) Cayley factor graphs of amenable groups, and that the Poisson point process on groups with Kazhdan's Property (T) admits no Cayley factor graphs. This gives examples of pmp countable Borel equivalence relations that cannot be generated by any free action of a countable group.