Geometry of infinite-dimensional groups

我们将应用无限维群的理论，特别是它们的几何、动力学和组合方面，来研究一些古老的公开问题，包括Connes的嵌入猜想和Gottschalk的结合性猜想。我们将通过考虑最近才引入但已经引起人们极大兴趣的离散群的中间类来实现这一点。这些群是作为一些具有丰富而有趣的附加结构的具体无限维群的子群得到的，但到目前为止很少受到关注。它们包括超线性群，它与著名的Connes嵌入猜想有关，其中每一次进展都将对许多领域产生重要的结果，如算子代数和自由概率。在这方面，我们将要研究的相应的无限维群是有限秩酉群与规范化的Hilbert-Schmidt距离的超积。另一个重要的类是SOFIC群，它是由Gromov引入的，作为在符号动力学中接近Gottschalk的附加性猜想的手段。SOFIC群是有限对称群关于正规Hamming度量的超积的子群。我们将研究这些群与其他密切相关的群之间的关系，包括具有Haagerup性质的群、初始从属群、在无穷远处从属的群等。我们基于无限维群的方法提出了对旧问题的新观点。

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.