#### DMCA

## Amenable actions and almost invariant sets

Venue: | Proc. Amer. Math. Soc |

Citations: | 18 - 2 self |

### Citations

66 |
Kazhdan’s property T, strong ergodicity and invariant means for ergodic group-actions, Ergodic Theory and
- Schmidt, Amenability
- 1981
(Show Context)
Citation Context ...ost invariant sets implies the existence of almost invariant vectors for the Koopman representation κ0 (look at the characteristic functions) but the converse may fail, as was first proved by Schmidt =-=[11]-=- (for another example, see Hjorth–Kechris [7, Theorem A3.2]). In fact, the existence of almost invariant sets depends only on the orbit equivalence relation which, in the ergodic case, is equivalent t... |

49 | A converse to Dye’s theorem,
- Hjorth
- 2005
(Show Context)
Citation Context ...erning orbit equivalence. Schmidt [11] showed that every non-amenable group Γ that does not have property (T) has at least two non-orbit equivalent, ergodic actions (this was extended later by Hjorth =-=[8]-=- to all non-amenable groups). The preceding result shows that if Γ is non-amenable but admits an action on X which is amenable and has infinite orbits (this class of groups is a subclass of non-proper... |

41 | Rigidity theorems for actions of product groups and countable Borel equivalence relations
- Hjorth, Kechris
(Show Context)
Citation Context ... on the orbit equivalence relation which, in the ergodic case, is equivalent to non E0-ergodicity (Jones–Schmidt [9]), while the existence of almost invariant vectors depends on the group action (see =-=[7]-=- again). Recall that E0 is the equivalence relation on 2 N defined by (xn) E0 (yn) ⇐⇒ ∃m∀n > m xn = yn. An equivalence relation E on a measure space (Y, µ) is E0-ergodic if for every Borel map f : Y →... |

26 |
Asymptotically invariant sequences and approximate finiteness
- Jones, Schmidt
- 1987
(Show Context)
Citation Context ...echris [7, Theorem A3.2]). In fact, the existence of almost invariant sets depends only on the orbit equivalence relation which, in the ergodic case, is equivalent to non E0-ergodicity (Jones–Schmidt =-=[9]-=-), while the existence of almost invariant vectors depends on the group action (see [7] again). Recall that E0 is the equivalence relation on 2 N defined by (xn) E0 (yn) ⇐⇒ ∃m∀n > m xn = yn. An equiva... |

20 | Amenable actions, free products and a fixed point property,
- Glasner, Monod
- 2007
(Show Context)
Citation Context ...tion; see Bédos–de la Harpe [1] for definitions and examples). Interestingly, free groups also admit transitive, faithful, amenable actions (van Douwen [3]). Y. Glasner and N. Monod in a recent paper =-=[5]-=- study the class of groups which admit transitive, faithful, amenable actions and give some history, references, and further examples. Grigorchuk–Nekrashevych [6] describe yet another example of faith... |

13 |
la Harpe: Moyennabilité intérieure des groupes: définitions et exemples, L’Enseignement
- Bédos, de
- 1986
(Show Context)
Citation Context ...actions with infinite orbits. Important examples are the non-amenable, inner amenable groups with infinite conjugacy classes (consider the action of Γ on Γ \ {1} by conjugation; see Bédos–de la Harpe =-=[1]-=- for definitions and examples). Interestingly, free groups also admit transitive, faithful, amenable actions (van Douwen [3]). Y. Glasner and N. Monod in a recent paper [5] study the class of groups w... |

10 | Nekrashevych, Amenable actions of non-amenable groups
- Grigorchuk, Yu
(Show Context)
Citation Context ...lasner and N. Monod in a recent paper [5] study the class of groups which admit transitive, faithful, amenable actions and give some history, references, and further examples. Grigorchuk–Nekrashevych =-=[6]-=- describe yet another example of faithful, transitive, amenable actions of free groups. On the other hand, every amenable action of a group with Kazhdan’s property (T) has a finite orbit. An action of... |

6 |
Douwen, Measures invariant under actions of F2
- van
- 1990
(Show Context)
Citation Context ... (consider the action of Γ on Γ \ {1} by conjugation; see Bédos–de la Harpe [1] for definitions and examples). Interestingly, free groups also admit transitive, faithful, amenable actions (van Douwen =-=[3]-=-). Y. Glasner and N. Monod in a recent paper [5] study the class of groups which admit transitive, faithful, amenable actions and give some history, references, and further examples. Grigorchuk–Nekras... |

4 |
Unitary representations and modular
- Kechris
(Show Context)
Citation Context ...he two concepts coincide. A relatively broad class of examples of measure preserving actions, studied by several authors (see the monograph Schmidt [12] for discussion and references and also Kechris =-=[10]-=-), consists of the actions by automorphisms on compact Polish groups (equipped with the Haar measure). The generalized Bernoulli shifts with a homogeneous base space M also fall into that class. Quest... |

3 |
Kazhdan’s property (T) (available at http://www.mmas.univ-metz.fr/∼bekka
- Bekka, Harpe, et al.
(Show Context)
Citation Context ...△ γ · F | < ɛ|F |. The following equivalences are well known and can be proved in exactly the same way as the corresponding ones for amenability of groups (see, for example, Bekka–de la Harpe–Valette =-=[2]-=-). Theorem 1.1. The following are equivalent for an action of Γ on X: (i) the action is amenable; (ii) the action satisfies the Følner condition; (iii) 1Γ ≺ λX. Clearly, all actions of amenable groups... |

2 |
Probability: theory and examples, Third
- Durrett
- 2005
(Show Context)
Citation Context ...he Central Limit Theorem for random variables several times and we find it convenient to employ probabilistic notation. For all necessary background in probability theory, a good reference is Durrett =-=[4]-=-. In this section, we will use P instead of µ to denote the measure on M X . Recall that a sequence ξk of random variables converges in distribution to ξ (written as ξk ⇒ ξ) if the distribution measur... |

1 |
systems of algebraic origin, Progress in Mathematics, vol. 128, Birkhäuser Verlag
- Dynamical
- 1995
(Show Context)
Citation Context ...ing to know whether there are other cases in which the two concepts coincide. A relatively broad class of examples of measure preserving actions, studied by several authors (see the monograph Schmidt =-=[12]-=- for discussion and references and also Kechris [10]), consists of the actions by automorphisms on compact Polish groups (equipped with the Haar measure). The generalized Bernoulli shifts with a homog... |