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**The absorption theorem for affable equivalence relations.**
*(English)*
Zbl 1166.37004

A. Connes, J. Feldman and B. Weiss [Ergodic Theory Dyn. Syst. 1, 431–450 (1981; Zbl 0491.28018)] proved that any free ergodic action of an amenable group by measure class preserving transformations is orbit equivalent to an action of the group of integers. This paper continues a series of results pursuing similar results in the setting of topological dynamics. In an earlier work, T. Giordano, I. Putnam and C. Skau [Ergodic Theory Dyn. Syst. 24, No. 2, 441–475 (2004; Zbl 1074.37010)] developed a sophisticated machinery to probe this kind of question for a countable amenable group acting freely and minimally on a compact metric zero-dimensional space \(X\). An equivalence relation with countable (possibly finite) equivalence classes is called étale if the projection \((x,y)\mapsto x\) is a local homeomorphism in a suitable topology on the relation \(R\subset X\times X\). An étale relation is called affable (AF-able, from the close relation to the notion of approximately finite dimensional in operator algebras) if it can be topologized so that it is the union of an increasing sequence of open subrelations each of which has the property that the complement of the diagonal in \(R\) is compact (that is, so that it is an AF equivalence relation). A basic example is that the orbit equivalence relation of an action of a locally finite countable group on a Cantor set is an AF equivalence relation. Various properties of and tests for this notion were developed.

In this paper the main result is extended significantly, by showing that a minimal AF equivalence relation \(R\) extended by modification on a thin closed subset \(Y\subset X\) is affable. This is already a substantial step forward if \(Y\) is required to be a finite set, and this rather technical result is a potent tool in the determination of orbit equivalence for \(\mathbb Z^d\) actions in particular, providing a key step in the recent work of the same four authors showing that any minimal \(\mathbb Z^2\) action on a Cantor set is orbit equivalent (to an AF equivalence relation and hence) to a minimal \(\mathbb Z\) action [J. Am. Math. Soc. 21, No. 3, 863–892 (2008; Zbl 1254.37012)].

In this paper the main result is extended significantly, by showing that a minimal AF equivalence relation \(R\) extended by modification on a thin closed subset \(Y\subset X\) is affable. This is already a substantial step forward if \(Y\) is required to be a finite set, and this rather technical result is a potent tool in the determination of orbit equivalence for \(\mathbb Z^d\) actions in particular, providing a key step in the recent work of the same four authors showing that any minimal \(\mathbb Z^2\) action on a Cantor set is orbit equivalent (to an AF equivalence relation and hence) to a minimal \(\mathbb Z\) action [J. Am. Math. Soc. 21, No. 3, 863–892 (2008; Zbl 1254.37012)].

Reviewer: Thomas Ward (Norwich)

### MSC:

37B10 | Symbolic dynamics |

22D25 | \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations |

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\textit{T. Giordano} et al., Ergodic Theory Dyn. Syst. 28, No. 5, 1509--1531 (2008; Zbl 1166.37004)

### References:

[1] | DOI: 10.1017/S014338570300066X · Zbl 1074.37010 |

[2] | Giordano, J. Reine Angew. Math. 469 pp 51– (1995) |

[3] | Molberg, Math. Scand. 99 pp 247– (2006) · Zbl 1137.37002 |

[4] | Paterson, Groupoids, Inverse Semigroups, and Their Operator Algebras (1999) |

[5] | DOI: 10.1090/S0894-0347-08-00595-X · Zbl 1254.37012 |

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