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**Cost of equivalence relations and groups.
(Coût des relations d’équivalence et des groupes.)**
*(French)*
Zbl 0939.28012

We study a new dynamical invariant for discrete groups: the cost. It is a real number in \(\{1-1/n\}\cup[1,\infty]\), bounded by the number of generators of the group.

It is dynamical because it relies on measure-preserving free actions on probability Borel spaces. In most cases, groups have fixed price, which implies that two freely acting groups which define the same orbit partition must have the same cost. It enables us to distinguish the orbit partitions of probability-preserving free actions of free groups of different ranks.

From the Ornstein-Weiss theorem, the cost of every infinite amenable group equals \(1\). We compute it in some other situations, including free products, free products with amalgamation and HNN-extensions over amenable groups and for direct product situations. For instance, the cost of the free group on \(n\) generators equals \(n\) and that of the fundamental group of a surface of genus \(g>0\) equals \(2g-1\). The cost is well behaved with respect to finite index subgroups. Namely, the quantities \(1\) minus the cost are related by multiplying by the index. We prove that each value in \(\{1- 1/n\}\cup [1,\infty)\) is achieved as the cost of a finitely generated group. At the end of the paper, we give a mercuriale, i.e., a list of costs of different groups.

The cost is in fact an invariant of ergodic measure-preserving countable equivalence relations and is defined using graphings. A graphing is a measurable way to provide (almost) every equivalence class (= orbit) with the structure of a connected graph. A treeing is an example of a graphing for which the orbits have the structure of simplicial trees. While every such relation admits graphings, not every relation admits a treeing: we prove that every free action of a cost \(1\) non-amenable group (and there are a lot) is not treeable. We prove that subrelations of treeable relations are treeable. We give examples of relations which cannot be produced by an action of any finitely generated group.

The cost of a relation which can be decomposed as a direct product is shown to be \(1\). We define the notion for a relation to be a free product or an HNN-extension and compute the cost for the resulting relation from the costs of the building blocks. For instance, the cost of a free product is the sum of the costs. The cost is also an invariant of the pairs von Neumann algebra/Cartan subalgebra.

It is dynamical because it relies on measure-preserving free actions on probability Borel spaces. In most cases, groups have fixed price, which implies that two freely acting groups which define the same orbit partition must have the same cost. It enables us to distinguish the orbit partitions of probability-preserving free actions of free groups of different ranks.

From the Ornstein-Weiss theorem, the cost of every infinite amenable group equals \(1\). We compute it in some other situations, including free products, free products with amalgamation and HNN-extensions over amenable groups and for direct product situations. For instance, the cost of the free group on \(n\) generators equals \(n\) and that of the fundamental group of a surface of genus \(g>0\) equals \(2g-1\). The cost is well behaved with respect to finite index subgroups. Namely, the quantities \(1\) minus the cost are related by multiplying by the index. We prove that each value in \(\{1- 1/n\}\cup [1,\infty)\) is achieved as the cost of a finitely generated group. At the end of the paper, we give a mercuriale, i.e., a list of costs of different groups.

The cost is in fact an invariant of ergodic measure-preserving countable equivalence relations and is defined using graphings. A graphing is a measurable way to provide (almost) every equivalence class (= orbit) with the structure of a connected graph. A treeing is an example of a graphing for which the orbits have the structure of simplicial trees. While every such relation admits graphings, not every relation admits a treeing: we prove that every free action of a cost \(1\) non-amenable group (and there are a lot) is not treeable. We prove that subrelations of treeable relations are treeable. We give examples of relations which cannot be produced by an action of any finitely generated group.

The cost of a relation which can be decomposed as a direct product is shown to be \(1\). We define the notion for a relation to be a free product or an HNN-extension and compute the cost for the resulting relation from the costs of the building blocks. For instance, the cost of a free product is the sum of the costs. The cost is also an invariant of the pairs von Neumann algebra/Cartan subalgebra.

Reviewer: Damien Gaboriau (Lyon)

### MSC:

28D15 | General groups of measure-preserving transformations |

37A20 | Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations |

28D20 | Entropy and other invariants |

20E15 | Chains and lattices of subgroups, subnormal subgroups |

20E06 | Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations |

46L10 | General theory of von Neumann algebras |

37A35 | Entropy and other invariants, isomorphism, classification in ergodic theory |