Ergodic Theoretic Methods in Group Homology : A Minicourse on L2-Betti Numbers in Group Theory. 1st ed. 2020

種類:

電子ブック

責任表示:

by Clara Löh

出版情報:

Cham : Springer International Publishing : Imprint: Springer, 2020

著者名:

• Löh, Clara.
• SpringerLink (Online service)

シリーズ名:

SpringerBriefs in Mathematics ;

ISBN:

9783030442200 [3030442209]    

注記:

0 Introduction -- 1 The von Neumann dimension -- 2 L2-Betti numbers -- 3 The residually finite view: Approximation -- 4 The dynamical view: Measured group theory -- 5 Invariant random subgroups -- 6 Simplicial volume -- A Quick reference -- Bibliography -- Symbols -- Index.
This book offers a concise introduction to ergodic methods in group homology, with a particular focus on the computation of L2-Betti numbers. Group homology integrates group actions into homological structure. Coefficients based on probability measure preserving actions combine ergodic theory and homology. An example of such an interaction is provided by L2-Betti numbers: these invariants can be understood in terms of group homology with coefficients related to the group von Neumann algebra, via approximation by finite index subgroups, or via dynamical systems. In this way, L2-Betti numbers lead to orbit/measure equivalence invariants and measured group theory helps to compute L2-Betti numbers. Similar methods apply also to compute the rank gradient/cost of groups as well as the simplicial volume of manifolds. This book introduces L2-Betti numbers of groups at an elementary level and then develops the ergodic point of view, emphasising the connection with approximation phenomena for homological gradient invar

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