A NONAMENABLE "FACTOR" OF A EUCLIDEAN SPACE

Answering a question of Benjamini, we present an isometry-invariant random partition of the Euclidean space R-d, 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 R-d 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's.