Orbit equivalence rigidity for ergodic actions of the mapping class group. (English) Zbl 1194.37012
Given an ergodic standard action \(\alpha\) of a discrete group, a is orbit equivalent super-rigid if: for any other ergodic standard action \(\beta\) of a discrete group which is weakly orbit equivalent to \(\alpha\), \(\alpha\) and \(\beta\) are virtually conjugate in a certain sense which is defined. In this paper, the author establishes orbit equivalence super-rigidity for certain ergodic, free measure preserving actions on a standard Borel probability space with a finite positive measure. This continues the work of A. Furman [Ann. Math. 150, 1083–1108 (1999; Zbl 0943.22012)] (based on work by R. J. Zimmer [Ergodic theory and semi-simple groups, Monogr. Math. 81, Birkhäuser Verlag, Basel (1984; Zbl 0571.58015)]), who showed orbit equivalent super-rigidity for some ergodic standard actions of a lattice in a simple Lie group of higher real rank.
MSC:
| 37A20 | Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations |
| 20F38 | Other groups related to topology or analysis |
| 37A35 | Entropy and other invariants, isomorphism, classification in ergodic theory |
| 37A15 | General groups of measure-preserving transformations and dynamical systems |
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