Measure equivalence rigidity of the mapping class group

By Y. Kida. 2006

70 citations recorded by the source corpus.

We show that the mapping class group of a compact orientable surface with higher complexity satisfies the following rigidity in the sense of measure equivalence: If the mapping class group is measure equivalent to a discrete group, then they are commensurable up to finite kernels. Moreover, we describe all locally compact second countable groups containing a lattice isomorphic to the mapping class group. We obtain similar results for finite direct products of mapping class groups.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/642149ed8f99f8b9397d98c3835d7c1e5d7d2d7b

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. S. Vaes
3. Sorin Popa
4. A. Furman
5. Xin Li
6. Damien Gaboriau
7. U. Bader
8. J. Peterson

In Mathematical Physics

1. Adrian Ioana
2. S. Vaes
3. Sorin Popa
4. A. Furman
5. Xin Li
6. Damien Gaboriau
7. J. Peterson
8. U. Bader

Citations in the corpus

• W*-superrigidity for Bernoulli actions of property (T) groups (2010, result): “Prior to these results, several large classes of OE– superrigid actions were found ([11], [37], [39], [21], [15], [22]).”
• Geometry of the mapping class groups III: Quasi-isometric rigidity (2005, unknown): “Kida [K06] recently showed an analogous rigidity result in the context of measure equivalence.”
• Integrable measure equivalence and rigidity of hyperbolic lattices (2010, background): “In all these results, the measure equivalence class of one of such groups turns out to be small and to consist of a list of ”obvious” examples obtained by simple modifications of the original group.”
• Classification and rigidity for von Neumann algebras (2012, background): “lli actions Γ y(X,µ) = (X0,µ0)Γ of many groups, including property (T) groups [Po05] and products of non-amenable groups [Po06a] (3) arbitrary free ergodic pmp actions of certain mapping class groups [Ki06] (4) free ergodic profinite actions of property (T) groups [Io08] Notethat theactionsΓ y(X,µ) f”
• Group measure space decomposition of II1 factors and W*-superrigidity (2009, background): “But while several large families of OE-superrigid actions have been discovered over the last ten years [19, 27, 31 , 32, 47, 49], unique Cartan decomposition proved to be much more challenging to establish, and the only existing results cover very particular group actions.”
• Continuous orbit equivalence rigidity (2015, background): “lished itself as a key notion. The idea is to find classes of actions for which orbit equivalence already implies conjugacy. Indeed, impressive orbit equivalence rigidity results have been obtained in [28, 7, 15, 16, 17, 14, 13]. Viewing continuous orbit equivalence as the topological analogue of orb”