Geometry of infinite-dimensional groups

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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.