Ergodic Theoretic Methods in Group Homology: A Minicourse on L2-Betti Numbers in Group Theory (SpringerBriefs in Mathematics)

This book offers a concise introduction to ergodic methods in group homology, with a particular focus on the computation of L 2 -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 L 2 -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, L 2 -Betti numbers lead to orbit/measure equivalence invariants and measured group theory helps to compute L 2 -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 L 2 -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. Contents 1 Introduction Overview of this minicourse Additional material 2 The von Neumann dimension 1.1 From the group ring to the group von Neumann algebra 1.1.1 The group ring 1.1.2 Hilbert modules 1.1.3 The group von Neumann algebra 1.2 The von Neumann dimension 1.E Exercises 3 L^2-Betti numbers 2.1 An elementary definition of L^2-Betti numbers 2.1.1 Finite type 2.1.2 L^2-Betti numbers of spaces 2.1.3 L^2-Betti numbers of groups 2.2 Basic computations 2.2.1 Basic properties 2.2.2 First examples 2.3 Variations and extensions 2.E Exercises 4 The residually finite view: Approximation 3.1 The approximation theorem 3.2 Proof of the approximation theorem 3.2.1 Reduction to kernels of self-adjoint operators 3.2.2 Reformulation via spectral measures 3.2.3 Weak convergence of spectral measures 3.2.4 Convergence at 0 3.3 Homological gradient invariants 3.3.1 Betti number gradients 3.3.2 Rank gradient 3.3.3 More gradients 3.E Exercises 5 The dynamical view: Measured group theory 4.1 Measured group theory 4.1.1 Standard actions 4.1.2 Measure/orbit equivalence 4.2 L^2-Betti numbers of equivalence relations 4.2.1 Measured equivalence relations 4.2.2 L^2-Betti numbers of equivalence relations 4.2.3 Comparison with L^2-Betti numbers of groups 4.2.4 Applications to orbit equivalence 4.2.5 Applications to L^2-Betti numbers of groups 4.3 Cost of groups 4.3.1 Rank gradients via cost 4.3.2 The cost estimate for the first L2-Betti number 4.3.3 Fixed price 4.E Exercises 6 Invariant random subgroups 5.1 Generalised approximation for lattices 5.1.1 Statement of the approximation theorem 5.1.2 Terminology 5.2 Two instructive examples 5.2.1 Lattices in SL(n, R) 5.2.2 Why doesn't it work in rank 1?! 5.3 Convergence via invariant random subgroups 5.3.1 Invariant random subgroups 5.3.2 Benjamini–Schramm convergence 5.3.3 The accumulation point 5.3.4 Reduction to Plancherel measures 5.3.5 Convergence of Plancherel measures 5.E Exercises 7 Simplicial volume 6.1 Simplicial volume 6.2 The residually finite view 6.3 The dynamical view 6.4 Basic proof techniques 6.4.1 The role of the profinite completion 6.4.2 Betti number estimates 6.4.3 The rank gradient/cost estimate 6.4.4 Amenable fundamental group 6.4.5 Hyperbolic 3-manifolds 6.4.6 Aspherical 3-manifolds 6.E Exercises A Quick reference A.1 Von Neumann algebras A.2 Weak convergence of measures A.3 Lattices Bibliography Symbols Index