Answering a question of Benjamini, we present an isometry-invariant random partition of the Euclidean space , , 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 as an isometry-invariant random partition of to bounded polyhedra, and also as an isometry-invariant random partition of to indistinguishable pieces. A new technique is developed to prove indistinguishability for certain constructions, connecting this notion to factor of IID’s.
"A nonamenable “factor” of a Euclidean space." Ann. Probab. 49 (3) 1427 - 1449, May 2021. https://doi.org/10.1214/20-AOP1485