A survey of Measured Group Theory

Abstract

The title refers to the area of research which studies infinite groups using measure-theoretic tools, and studies the restrictions that group structure imposes on ergodic theory of their actions. The paper is a survey of recent developments focused on the notion of Measure Equivalence between groups, and Orbit Equivalence between group actions. We discuss known invariants and classification results (rigidity) in both areas.

A survey for the Proceedings of a Conference honoring Robert Zimmer's 60th birthday. 61 pages. Referees comments are incorporated, in particular sections 3.1.2 (cost) and 3.1.4 (Cowling Haagerup invariant) were updated. Some new references added

References in corpus (14)

• The mapping class group from the viewpoint of measure equivalence theory
• Groups acting on manifolds: around the Zimmer program
• Orbit inequivalent actions of non-amenable groups
• Mapping Class Groups do not have Kazhdan's Property (T)
• Measure Equivalence Rigidity and Bi-exactness of Groups
• Outer automorphism groups of some ergodic equivalence relations
• A spectral sequence to compute L2-Betti numbers of groups and groupoids
• Rank gradient, cost of groups and the rank versus Heegaard genus problem
• Free products, Orbit Equivalence and Measure Equivalence Rigidity
• Cocycle and orbit superrigidity for lattices in SL(n,R) acting on homogeneous spaces
• On a class of II_1 factors with at most one Cartan subalgebra II
• Superrigidity, Weyl groups, and actions on the Circle
• The intrinsic asymmetry and inhomogeneity of Teichmuller space
• Non-Orbit Equivalent Actions of ${\mathbb{F}}_n$

Cited by in corpus (14)

• OE and W* superrigidity results for actions by surface braid groups
• Examples of amalgamated free products and coupling rigidity
• A new lattice invariant for lattices in totally disconnected locally compact groups
• Boundary amenability and measure equivalence rigidity among two-dimensional Artin groups of hyperbolic type
• Superrigidity from Chevalley groups into acylindrically hyperbolic groups via quasi-cocycles
• Baumslag-Solitar groups, relative profinite completions and measure equivalence rigidity
• Measure equivalence rigidity of $\mathrm{Out}(F_N)$
• Orbit equivalence, coinduced actions and free products
• Superrigidity, arithmeticity, normal subgroups: results, ramifications and directions
• Stable orbit equivalence of Bernoulli shifts over free groups
• Point processes, cost, and the growth of rank in locally compact groups
• Weak amenability of free products of hyperbolic and amenable groups
• Measure equivalence rigidity of the handlebody groups
• The Palm groupoid of a point process and factor graphs on amenable and Property (T) groups