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Trees and amenable equivalence relations

Published online by Cambridge University Press:  19 September 2008

Scot Adams
Scot Adams
Affiliation:
Department of Mathematics, Stanford University, Stanford, California 94305, USA
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Abstract

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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 makin 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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 10 , Issue 1 , March 1990 , pp. 1 - 14
Copyright
Copyright © Cambridge University Press 1990

References

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