The Brin–Thompson groups sV are of type F∞
The Brin–Thompson groups sV are of type F∞
复制标题
Brin–Thompson 群 sV 属于 F∞ 类型
DOI：
--
复制
发表时间：
2012
期刊：
影响因子：
--
通讯作者：
M. C. B. Zaremsky
中科院分区：
文献类型：
--
作者：
Martin G. Fluch;Marco Marschler;Stefan Witzel;M. C. B. Zaremsky
我们证明了布林-汤普森群sV，也称为高维汤普森群，对所有s2 N都是F1型。这一结果以前显示为s 3，通过考虑的作用sV上的一个自然相关的空间。我们的关键步骤是用一个更容易分析的子空间sX来代替这个空间。回想一下，一个群是F1型的，如果它允许一个分类空间在每个维度上有100个单元格。F1型群的著名例子包括汤普森群F、T和V。Brin [2004; 2005]介绍了V的一些推广，并证明是简单的。我们用1VD V表示这些群sV，对于s2 N。这些群通常被称为高维汤普森群或布林-汤普森群。已知所有sV群均为双列型[Hennig and Matucci 2012]，Kochloukova、Martinez-Perez和Nucinkis [Kochloukova et al. 2013]表明2 V和3V属于F1型。我们证明了这个结果可以推广到所有的维度。主要定理布林-汤普森群sV对所有s都是F1型。
We prove that the Brin‐Thompson groups sV , also called higher-dimensional Thompson’s groups, are of type F1 for all s2 N. This result was previously shown for s 3, by considering the action of sV on a naturally associated space. Our key step is to replace this space by a subspace sX that is easier to analyze. Recall that a group is of type F1 if it admits a classifying space with finitely many cells in each dimension. Well-known examples of groups of type F1 include Thompson’s groups F, T , and V . Some generalizations of V were introduced by Brin [2004; 2005] and shown to be simple. We denote these groups sV , for s2 N, with 1VD V . These groups are usually termed higher-dimensional Thompson’s groups or Brin‐Thompson groups. All of the groups sV are known to be finitely presented [Hennig and Matucci 2012], and Kochloukova, Martinez-Perez, and Nucinkis [Kochloukova et al. 2013] showed that 2V and 3V are of type F1. We prove that this result extends to all dimensions. Main Theorem. The Brin‐Thompson group sV is of type F1 for all s.