# Tree and grid factors of general point processes

Adam Timar

Arxiv ID: 0909.1092•Last updated: 5/11/2020

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.

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