A nonamenable “factor” of a Euclidean space

Abstract

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