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100   $a20200314d2020      u|         0
101 0 $aeng
135   $aurnn|008mamaa
181   $ctxt$2rdacontent
182   $cc$2rdamedia
183   $acr$2rdacarrier
200 10$aErgodic Theoretic Methods in Group Homology $eA Minicourse on L2-Betti Numbers in Group Theory /$fby Clara Löh
205   $a1st ed. 2020.
210  1$aCham :$cSpringer International Publishing :$cImprint: Springer,$d2020.
215   $a1 online resource (IX, 114 p. 1 illus.) 
225 1 $aSpringerBriefs in Mathematics,$x2191-8198
311   $a3-030-44219-5 
327   $a0 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.
330   $aThis book offers a concise introduction to ergodic methods in group homology, with a particular focus on the computation of L2-Betti numbers. Group homology integrates group actions into homological structure. Coefficients based on probability measure preserving actions combine ergodic theory and homology. An example of such an interaction is provided by L2-Betti numbers: these invariants can be understood in terms of group homology with coefficients related to the group von Neumann algebra, via approximation by finite index subgroups, or via dynamical systems. In this way, L2-Betti numbers lead to orbit/measure equivalence invariants and measured group theory helps to compute L2-Betti numbers. Similar methods apply also to compute the rank gradient/cost of groups as well as the simplicial volume of manifolds. This book introduces L2-Betti numbers of groups at an elementary level and then develops the ergodic point of view, emphasising the connection with approximation phenomena for homological gradient invariants of groups and spaces. The text is an extended version of the lecture notes for a minicourse at the MSRI summer graduate school ?Random and arithmetic structures in topology? and thus accessible to the graduate or advanced undergraduate students. Many examples and exercises illustrate the material.
410  0$aSpringerBriefs in Mathematics,$x2191-8198
606   $aGroup theory
606   $aAlgebraic topology
606   $aDynamics
606   $aErgodic theory
606   $aVibration
606   $aDynamical systems
606   $aTopological groups
606   $aLie groups
606   $aGroup Theory and Generalizations$3https://scigraph.springernature.com/ontologies/product-market-codes/M11078
606   $aAlgebraic Topology$3https://scigraph.springernature.com/ontologies/product-market-codes/M28019
606   $aDynamical Systems and Ergodic Theory$3https://scigraph.springernature.com/ontologies/product-market-codes/M1204X
606   $aVibration, Dynamical Systems, Control$3https://scigraph.springernature.com/ontologies/product-market-codes/T15036
606   $aTopological Groups, Lie Groups$3https://scigraph.springernature.com/ontologies/product-market-codes/M11132
615  0$aGroup theory.
615  0$aAlgebraic topology.
615  0$aDynamics.
615  0$aErgodic theory.
615  0$aVibration.
615  0$aDynamical systems.
615  0$aTopological groups.
615  0$aLie groups.
615 14$aGroup Theory and Generalizations.
615 24$aAlgebraic Topology.
615 24$aDynamical Systems and Ergodic Theory.
615 24$aVibration, Dynamical Systems, Control.
615 24$aTopological Groups, Lie Groups.
676   $a512.2
700   $aLöh$b Clara$4aut$4http://id.loc.gov/vocabulary/relators/aut$0767630
801  0$bMiAaPQ
801  1$bMiAaPQ
801  2$bMiAaPQ
906   $aBOOK
912   $a9910484190903321
996   $aErgodic Theoretic Methods in Group Homology$91914842
997   $aUNINA