Orbit Equivalence and Measured Group Theory

@article{Gaboriau2010OrbitEA,
  title={Orbit Equivalence and Measured Group Theory},
  author={Damien Gaboriau},
  journal={arXiv: Group Theory},
  year={2010}
}
  • D. Gaboriau
  • Published 1 September 2010
  • Mathematics
  • arXiv: Group Theory
We give a survey of various recent developments in orbit equivalence and measured group theory. This subject aims at studying infinite countable groups through their measure preserving actions. 
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70 Citations
Highly Influential Citations
1
Background Citations
18
Methods Citations
4
Results Citations
3

70 Citations

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References

SHOWING 1-10 OF 238 REFERENCES
Orbit inequivalent actions for groups containing a copy of F 2
We prove that if a countable group contains a copy of F2, then it admits uncountably many non orbit equivalent actions.
A SURVEY OF MEASURED GROUP THEORY
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 er- godic theory of their actions.…
Orbit equivalence rigidity for ergodic actions of the mapping class group
We establish orbit equivalence rigidity for any ergodic, essentially free and measure-preserving action on a standard Borel space with a finite positive measure of the mapping class group for a…
I. Orbit Equivalence
1. Group Actions and Equivalence Relations 2. Invariant Measures 3. Ergodicity 4. Isomorphism and Orbit Equivalence
Topics in orbit equivalence
Preface.- I. Orbit Equivalence.- II. Amenability and Hyperfiniteness.- III. Costs of Equivalence Relations and Groups.- References.- Index.
Property T and asymptotically invariant sequences
A countable group Γ has property T of Kazhdan if and only if no measure preserving ergodic action of Γ has non-trivial asymptotically invariant sets.
On the cost of generating an equivalence relation
  • G. Levitt
  • Mathematics
    Ergodic Theory and Dynamical Systems
  • 1995
Abstract Given a measure-preserving equivalence relation R with countable classes, we study relations between the properties of R and metric invariants. We give applications to pseudogroups of…
Countable Borel Equivalence Relations
This paper develops the foundations of the descriptive set theory of countable Borel equivalence relations on Polish spaces with particular emphasis on the study of hyper-finite, amenable, treeable…
Deformation and rigidity for group actions and von Neumann algebras
We present some recent rigidity results for von Neumann algebras (II1 factors) and equivalence relations arising from measure preserving actions of groups on probability spaces which satisfy a…
Rigidity for von Neumann algebras and their invariants
We give a survey of recent classification results for crossed product von Neumann algebras arising from measure preserving group actions on probability spaces. This includes II_1 factors with…
...
...