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，来研究一些古老的开放问题，包括Connes的嵌入猜想和Gottschalk的上合猜想。我们将通过考虑离散群的中间类来做到这一点，离散群是最近才引入的，但已经引起了很大的兴趣。这些群是一些具体的无限维群的子群，它们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cones嵌入猜想有关，其中的每一个进展都会对许多领域产生重要影响，比如算子代数和自由概率。我们在这里要研究的无限维群是有限秩酉群与归一化希尔伯特-施密特距离的超积。另一个重要的类别是sofic群，由Gromov引入，作为在符号动力学中接近Gottschalk的上合猜想的手段。Sofic群是有限对称群关于归一化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、初始子可服从群、无穷可服从群等。我们基于无限维群的方法为老问题提出了一种新的观点。

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.