Answering a question of Benjamini, we present an isometry-invariant random
partition of the Euclidean space Rd, 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
Rd as an isometry-invariant random partition of Rd to
bounded polyhedra, and also as an isometry-invariant random partition of
Rd to indistinguishable pieces. A new technique is developed to
prove indistinguishability for certain constructions, connecting this notion to
factor of iid's.Comment: 23 pages, 4 figure