Orbit equivalence rigidity and bounded cohomology

By Nicolas Monod, Yehuda Shalom. 2006

227 citations recorded by the source corpus.

We establish new results and introduce new methods in the theory of measurable orbit equivalence, using bounded cohomology of group representations. Our rigidity statements hold for a wide (uncountable) class of groups arising from negative curvature geometry. Amongst our applications are (a) measurable Mostow-type rigidity theorems for products of negatively curved groups; (b) prime factorization results for measure equivalence; (c) superrigidity for orbit equivalence; (d) the first examples of continua of type II1 equivalence relations with trivial outer automorphism group that are mutually not stably isomorphic.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a2908b75d5a41a65611e1ba008f8db8729a08bf3

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Adrian Ioana
2. A. Furman
3. S. Vaes
4. Sorin Popa
5. Damien Gaboriau
6. Denis V. Osin
7. Thomas Sinclair
8. Y. Kida

In Mathematical Physics

1. Adrian Ioana
2. A. Furman
3. S. Vaes
4. Sorin Popa
5. Damien Gaboriau
6. Thomas Sinclair
7. Denis V. Osin
8. I. Chifan

Citations in the corpus

• Acylindrically hyperbolic groups (2013, background): “It opens the door for Monod-Shalom rigidity theory for group actions on spaces with measure [39].”
• Acylindrical hyperbolicity of groups acting on trees (2013, unknown): “Thus C /∈ Creg ∪Dreg whenever C is of type (II) or (III).”
• Continuous orbit equivalence rigidity (2015, background): “Indeed, impressive orbit equivalence rigidity results have been obtained in [28, 7, 15, 16, 17, 14, 13].”
• Measure equivalence rigidity of the mapping class group (2006, background): “Note that if κ(M) 0 and M 6= M1,0, then Γ(M) is finite.”
• ORBIT INEQUIVALENT ACTIONS OF NON-AMENABLE GROUPS (2007, background): “question of which groups admit continuum many orbit inequivalent actions has also been implicitly or explicitly considered as well as answered for classes of certain groups in the papers Monod-Shalom [MS06], Popa [Pop06], Kechris [Kec], Tornquist [To¨r05], Fernos [Fer06]. The main goal of this paper”
• Rigidity results for Bernoulli actions and their von Neumann algebras (after Sorin Popa) (2006, background): “the other side. Note however that Furman’s theorem nevertheless depends on Zimmer’s cocycle rigidity theorem. We also mention the orbit equivalence superrigidity theorems obtained by Monod and Shalom [39] for certain actions of direct products of hyperbolic groups. An excellent overview oforbitequiv”
• Rigidity Theorems For Actions Of Product Groups And Countable Borel Equivalence Relations (2005, methodology): “Addendum 1. After the first draft of this paper was completed, we received two recent preprints by N. Monod and Y. Shalom, [MS02, MS02a ], which also contain several rigidity results about product groups, in the measure theoretic context, obtained by using the techniques of bounded cohomology.”
• Bounded cohomology and isometry groups of hyperbolic spaces (2005, methodology): “ndard infinite measure Borel space with finite measure fundamental domains. Measure equivalence defines an equivalence relation for countable groups [Fu99a]. It follows from recent work Monod and Shalom [MS05] that two measure equivalent countable groups have isomorphic second bounded cohomology groups”
• Amalgamated free products of weakly rigid factors and calculation of their symmetry groups (2005, methodology): “lence relation R 1, 1 having trivial outer automorphism group are constructed in [Ge2], [Fu2] and [ MoS ], and we construct some more, using the Monod‐Shalom rigidity theorem [MoS].”