Titlebook: Ergodic Theoretic Methods in Group Homology; A Minicourse on L2-B Clara L?h Book 2020 The Author(s), under exclusive license to Springer Na

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Book 2020y level and thendevelops the ergodic point of view, emphasising the connection with approximation phenomena for homological gradient invariants of groups and spaces. The text is an extended version of the lecture notes for a minicourse at the MSRI summer graduate school “Random and arithmetic struct

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Teresa Semedo-Lemsaddek,Maria Jo?o FraquezaThis minicourse gives a brief introduction to ergodic theoretic methods in group homology. By now, this is a vast subject. In the present course, we will focus on .-Betti numbers. This chapter gives an overview of the course and lists additional material related to .-Betti numbers of groups and related fields.

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Hely H?ggman,Suvi Sutela,Matthias FladungWe will now use the von Neumann dimension to define .-Betti numbers of groups and spaces. We will study basic properties of .-Betti numbers, and we will compute some simple examples. Furthermore, we will briefly outline generalisations of .-Betti numbers beyond our elementary approach.

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Introduction,This minicourse gives a brief introduction to ergodic theoretic methods in group homology. By now, this is a vast subject. In the present course, we will focus on .-Betti numbers. This chapter gives an overview of the course and lists additional material related to .-Betti numbers of groups and related fields.

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Clara L?hMakes recent developments on L2-Betti numbers of groups and related invariants easily accessible to advanced students and researchers.Explains the rich interplay between the residually finite approach

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Redouane Choukr-Allah,Ragab Ragabrings. We explain the (spectral) proof of this approximation theorem and briefly discuss the relation with other (homological) gradient invariants. This residually finite view will be complemented by the dynamical view in Chap. . and the approximation theorems for lattices in Chap. ..

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The effect of compatible solutes on proteinscomputed in terms of probability measure preserving actions..We will first recall basic terminology from measured group theory. Then we will study .-Betti numbers of standard equivalence relations. Moreover, we will discuss cost and its relation with the first .-Betti number and rank gradients.