Tree and grid factors of general point processes

Abstract

We study isomorphism invariant point processes of $\mathbb{R}^d$ whose groups of symmetries are almost surely trivial. We define a 1-ended, locally finite tree factor on the points of the process, that is, a mapping of the point configuration to a graph on it that is measurable and equivariant with the point process. This answers a question of Holroyd and Peres. The tree will be used to construct a factor isomorphic to $\Z^n$. This perhaps surprising result (that any $d$ and $n$ works) solves a problem by Steve Evans. The construction, based on a connected clumping with $2^i$ vertices in each clump of the $i$'th partition, can be used to define various other factors.