The Palm groupoid of a point process and factor graphs on amenable and Property (T) groups

TL;DR

This paper introduces a new groupoid framework for invariant point processes on groups, revealing that amenable groups admit Cayley factor graphs while Property (T) groups do not, impacting the understanding of group actions.

Contribution

It defines a probability measure preserving groupoid associated to point processes and uses it to characterize the existence of Cayley factor graphs on different groups.

Findings

Amenable groups admit all Cayley factor graphs.

Property (T) groups admit no Cayley factor graphs.

Provides examples of equivalence relations not generated by free group actions.

Abstract

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.