@article{be12b9f631c84235b8be1321fbca16a4,
title = "A NONAMENABLE “FACTOR” OF A EUCLIDEAN SPACE",
abstract = "Answering a question of Benjamini, we present an isometry-invariant random partition of the Euclidean space Rd, d ≥ 3, into infinite connected indistinguishable pieces, such that the adjacency graph defined on the pieces is the 3-regular infinite tree. Along the way, it is proved that any finitely generated one-ended amenable Cayley graph can be represented in R d as an isometry-invariant random partition of Rd to bounded polyhedra, and also as an isometry-invariant random partition of R d to indistinguishable pieces. A new technique is developed to prove indistinguishability for certain constructions, connecting this notion to factor of IID{\textquoteright}s.",
keywords = "Random tiling, factor of IID, indistinguishability, isometry-invariant tiling",
author = "{\'A}d{\'a}m Tim{\'a}r",
note = "Funding Information: This work was started at the Bernoulli Center (CIB) conference “Statistical physics on transitive graphs.” I would like to thank Itai Benjamini, Dorottya Beringer, Damien Gaboriau, Russ Lyons, G{\'a}bor Pete, Mikael De La Salle and Romain Tessera for inspiring conversations, and a referee for useful comments. The author was supported by a Marie Curie Intra European Fellowship within the 7th European Community Framework Programme, by the Hungarian National Research, Development and Innovation Office, NKFIH Grant K109684, and by Grant LP 2016-5 of the Hungarian Academy of Sciences. After finishing but before publishing this paper, the author started working part-time at the University of Iceland. Funding Information: The author was supported by a Marie Curie Intra European Fellowship within the 7th European Community Framework Programme, by the Hungarian National Research, Development and Innovation Office, NKFIH Grant K109684, and by Grant LP 2016-5 of the Hungarian Academy of Sciences. After finishing but before publishing this paper, the author started working part-time at the University of Iceland. Publisher Copyright: {\textcopyright} Institute of Mathematical Statistics, 2021",
year = "2021",
month = may,
doi = "10.1214/20-AOP1485",
language = "English",
volume = "49",
pages = "1427--1449",
journal = "Annals of Probability",
issn = "0091-1798",
publisher = "Institute of Mathematical Statistics",
number = "3",
}