Coût des relations d’équivalence et des groupes
Coût des relations d’équivalence et des groupes
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等价关系和群体关系
DOI：
10.1007/s002229900019
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发表时间：
2000
影响因子：
3.1
通讯作者：
Damien Gaboriau
中科院分区：
文献类型：
--
作者：
Damien Gaboriau
摘要：我们研究了离散群的一个新的动力学不变量：代价。它是{1−1/n}<$[1，∞]中的真实的，以群的生成元个数为界，并且对于有限指数子群表现良好。也就是说，数量1减去成本通过乘以指数来关联。每个无限服从群的成本等于1。我们计算它在一些其他的情况下，包括自由产品，自由产品与合并和HNN扩展顺从组和直积的情况。例如，n个生成器上的空闲组的成本等于n。我们证明了每一个可能的有限值的成本是由一个可生成的群。它是动态的，因为它依赖于概率Borel空间上的保测度自由作用。在大多数情况下，群有固定的价格，这意味着定义相同轨道划分的两个自由行动群必须有相同的成本。它使我们能够区分不同秩的自由群的概率保持自由作用的轨道划分。在文章的最后，我们给出了一个mercuriale，即不同群体的成本列表。成本实际上是遍历测度保持等价关系的不变量，并使用图形定义。树是一种可测量的方式，为每个等价类（=轨道）提供一个单纯树的结构，这是一个图形的例子。不是每个关系承认树：我们证明了每一个自由行动的成本1不服从组是不树，但我们证明了子关系树的关系。我们给出的例子的关系，不能产生的行动，任何一个生成的组。可以分解为直积的关系的成本被示为1。我们将关系定义为自由积或HNN-扩展，并从构建块的成本中计算结果关系的成本。代价也是vonNeumann代数/Cartan子代数对的不变量。
Abstract.We study a new dynamical invariant for dicrete groups: the cost. It is a real number in {1−1/n}∪[1,∞], bounded by the number of generators of the group, and it is well behaved with respect to finite index subgroups. Namely, the quantities 1 minus the cost are related by multiplying by the index. 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. We prove that each possible finite value of the cost is achieved by a finitely generated 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. 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 equivalence relations and is defined using graphings. A treeing is a measurable way to provide every equivalence class (=orbit) with the structure of a simplicial tree, this an example of graphing. Not every relation admits a treeing: we prove that every free action of a cost 1 non-amenable group is not treeable, but 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. The cost is also an invariant of the pairs von Neumann algebra/Cartan subalgebra.