A nonamenable "factor" of a Euclidean space

Abstract

Answering a question of Benjamini, we present an isometry-invariant random partition of the Euclidean space ${\mathbb{R}}^d$, $d\ge 3$, 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 ${\mathbb{R}}^d$ as an isometry-invariant random partition of ${\mathbb{R}}^d$ to bounded polyhedra, and also as an isometry-invariant random partition of ${\mathbb{R}}^d$ to indistinguishable pieces. A new technique is developed to prove indistinguishability for certain constructions, connecting this notion to factor of iid's.

23 pages, 4 figures