论文标题
作用于cantor套件的群体的有界同谋
Bounded Cohomology of Groups acting on Cantor sets
论文作者
论文摘要
我们研究了作用于Cantor集合的某些群体的有界共同体。更具体地说，我们考虑了Cantor Set的全部同构和Thompson的Group $ V $。我们证明，这两个群体都是无环的，这是有界的共同体，具有微不足道的实际系数在正度上消失。将此结果与已经建立的$ \ mathbb {z} $ - 汤普森组$ v $的无效性，将使$ v $成为有限生成的组的第一个例子，实际上是一组$ f_ \ infty $的第一个例子，这是普遍限制的acyclicclic。在证明有界的无限性之前，我们收集了所考虑的各组及其某些亚组的各种特性。结果，有限的无环的证明将相对较短。事实证明，处理这些组的方法非常相似。这表明可能有一种统一的方法，这意味着在包括讨论的群体在内的较大类别的群组的有限范围的无环性。
We study the bounded cohomology of certain groups acting on the Cantor set. More specifically, we consider the full group of homeomorphisms of the Cantor set as well as Thompson's group $V$. We prove that both of these groups are boundedly acyclic, that is the bounded cohomology with trivial real coefficients vanishes in positive degrees. Combining this result with the already established $\mathbb{Z}$-acyclicity of Thompson's group $V$, will make $V$ the first example of a finitely generated group, in fact the first example of a group of type $F_\infty$, which is universally boundedly acyclic. Before proving bounded acyclicity, we gather various properties of the groups under consideration and certain subgroups thereof. As a consequence the proofs of bounded acyclicity will be relatively short. It will turn out that the approaches to handle these groups are very similar. This suggests that there could be a unifying approach which would imply the bounded acyclicity of a larger class of groups acting on the Cantor set, including the discussed ones.
