Geometry of infinite-dimensional groups

无限维群的几何

• 批准号: 261450-2007
• 负责人: Pestov, Vladimir
• 金额: $ 1.46万
• 依托单位: University of Ottawa
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2007
• 资助国家: 加拿大
• 起止时间: 2007-01-01 至 2008-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=366515
• 关键词: Geometry; infinite; dimensional; groups

We will apply theory of infinite-dimensional groups, especially their geometric, dynamical, and combinatorial aspects, to studying some old open problems, including Connes' Embedding Conjecture and Gottschalk's Surjunctivity Conjecture.We will be doing this by means of considering intermediate classes of discrete groups, introduced only recently but already causing a significant interest. These groups are obtained as subgroups of some concrete infinite-dimensional groups equipped with rich and interesting additional structures but receiving little attention up until now. They include hyperlinear groups, which are linked to the famous Connes' Embedding Conjecture, and where every advance will have important consequences for a number of areas, such as operator algebras and free probability. The corresponding infinite-dimensional group that we will be studying in this connection is the ultraproduct of the unitary groups of finite rank with the normalised Hilbert-Schmidt distance. Another important class is that of sofic groups, introduced by Gromov as means to approach Gottschalk's surjunctivity conjecture in symbolic dynamics. Sofic groups are subgroups of the ultraproduct of finite symmetric groups with regard to the normalized Hamming metric. We will study the relationships between these and other closely related classes of groups, including groups with Haagerup property, initially subamenable groups, groups amenable at infinity, and others. Our approach based on infinite-dimensional groups proposes a new viewpoint of the old problems.

我们将应用无穷维群的理论，特别是它们的几何、动力学和组合方面，来研究一些老的开放问题，包括Connes的嵌入猜想和Gottschalk的Surjunctivity猜想。我们将通过考虑离散群的中间类来做到这一点，这些中间类最近才介绍，但已经引起了很大的兴趣。这些群是作为某些具体的无限维群的子群而得到的，这些群具有丰富而有趣的附加结构，但迄今为止很少受到关注。它们包括超线性群，这与著名的康纳斯嵌入猜想有关，并且其中的每一个进步都将对许多领域产生重要影响，例如算子代数和自由概率。相应的无限维群，我们将在这方面的研究是超产品的酉群有限秩与正规化希尔伯特-施密特距离。另一个重要的类别是sofic群，介绍了格罗莫夫作为手段，以接近戈特沙尔克的surjunctivity猜想在符号动力学。Sofic群是有限对称群的超积关于正规化汉明度量的子群。我们将研究这些和其他密切相关的群类之间的关系，包括具有Haagerup性质的群，初始服从的群，无穷服从的群等。我们的方法基于无限维群提出了一个新的观点的老问题。