A NONAMENABLE FACTOR OF A EUCLIDEAN SPACE

• 成果类型: Article
• 署名作者: Timar, Adam
• 署名单位: HUN-REN; HUN-REN Alfred Renyi Institute of Mathematics
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/20-AOP1485
• 发表日期: 2021
• 页码: 1427-1449
• 关键词: percolation; indistinguishability

摘要

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.