Final Report Summary - ANALYTIC (ANALYTIC PROPERTIES OF INFINITE GROUPS:limits, curvature, and randomness) The overall goal of this project is to develop new concepts and techniques in geometric and asymptotic group theory for a systematic study of the analytic properties of discrete groups. Our research results have greatly contributed to establishing a new subject, the Analytic group theory, which generates growing interest worldwide both from experts and young researchers working not only in geometry and analysis of infinite groups but also in neighboring areas such as dynamical systems, operator algebra, ergodic theory, metric geometry, algebraic topology. Main outcomes of the project are the following. (1) The first constructions of graphs and groups with amazing large-scale properties. Examples are: the true coarse expander, that is, a regular graph which is not coarsely amenable but which is coarsely embeddable into Hilbert space; the Haagerup monster group, that is, a finitely generated group with the Haagerup property which is not coarsely amenable; a regular graph and, moreover, a finitely generated group which do not coarsely embed into Hilbert space and yet admit no weakly embedded expander. Thus, we produce new graphs and groups with a strong contrast behavior between metric properties and analytic attributes. (2) A first thorough study of analytic properties of direct limits of Gromov hyperbolic groups. For instance, we deal with the Haagerup property (= Gromov's a-T-menability) and the Kazhdan property (T), the Rapid Decay property and the Metric Approximation property, the C*-exactness and the C*-simplicity of infinitely presented classical and/or graphical small cancellation groups. In proving analytic results, we develop numerous novel tools to understand in detail geometric and combinatorial properties of these groups. Our results and methods apply both to fundamental examples, such as the Gromov monster and the Haagerup monster groups, and beyond.(3) We introduce and study metric approximations of algebras and the concept of stable metric approximations of groups. We provide a general framework, using metric ultraproducts, to analyze the stability of metric approximations of groups and algebras. We explore the “robustness” of metric approximations and indicate a way to the construction of a non-sofic group, a major open problem in the area. Our results are first in this direction and open a new strategy towards concrete examples and counterexamples in the currently fast-developing theory of sofic and hyperlinear groups.
