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Trees and amenable equivalence relations. (English) Zbl 0667.28003

Let R be a Borel equivalence relation with countable equivalence classes on a measure space M. Intuitively, a ‘treeing’ of R is a measurably- varying way of making each equivalence class into the vertices of a tree. We make this definition rigorous. We prove that if each equivalence class becomes a tree with polynomial growth, then the equivalence relation is amenable. We prove that if the equivalence relation is finite measure- preserving and amenable, then almost every tree (i.e., equivalence class) must have one or two ends.
Reviewer: S.Adams

MSC:

28D05 Measure-preserving transformations
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[1] DOI: 10.2307/1997924 · Zbl 0369.22009 · doi:10.2307/1997924
[2] DOI: 10.1016/0001-8708(78)90114-7 · Zbl 0392.28023 · doi:10.1016/0001-8708(78)90114-7
[3] Connes, Ergod. Th. Dynam. Sys. 1 pp 431– (1981)
[4] Arveson, An Invitation to C*-algebras (1976) · Zbl 0344.46123 · doi:10.1007/978-1-4612-6371-5
[5] Zimmer, Ergodic Theory and Semisimple Groups (1984) · doi:10.1007/978-1-4684-9488-4
[6] DOI: 10.2307/2374302 · Zbl 0527.57015 · doi:10.2307/2374302
[7] DOI: 10.2307/1997925 · Zbl 0369.22010 · doi:10.2307/1997925
[8] Zimmer, Ill. J. Math. 20 pp 373– (1976)
[9] DOI: 10.1007/BF01390162 · Zbl 0361.46061 · doi:10.1007/BF01390162
[10] Serre, Trees (1980) · Zbl 0548.20018 · doi:10.1007/978-3-642-61856-7
[11] DOI: 10.1016/0001-8708(71)90018-1 · Zbl 0216.14902 · doi:10.1016/0001-8708(71)90018-1
[12] DOI: 10.1007/BF01361167 · Zbl 0178.38802 · doi:10.1007/BF01361167
[13] Zimmer, Ergod. Th. & Dynam. Sys. 1 pp 237– (1981)
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