Answering a question of Benjamini, we present an isometry-invariant random partition of the euclidean space R^3 into infinite connected indinstinguishable 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 amenable Cayley graph (or more generally, amenable unimodular random graph) can be represented in R^3 as an isometry-invariant random collection of polyhedral domains (tiles). A new technique is developed to prove indistinguishability for certain constructions, connecting this notion to factor of iid's. 查看全文>>