scholarly journals Tiling Deformations, Cohomology, and Orbit Equivalence of Tiling Spaces

2018 ◽  
Vol 19 (10) ◽  
pp. 3053-3088 ◽  
Author(s):  
Antoine Julien ◽  
Lorenzo Sadun
Keyword(s):  
Orbit Equivalence ◽  
Tiling Spaces

scholarly journals On Polish groups admitting non-essentially countable actions

2020 ◽  
pp. 1-15
Author(s):  
ALEXANDER S. KECHRIS ◽  
MACIEJ MALICKI ◽  
ARISTOTELIS PANAGIOTOPOULOS ◽  
JOSEPH ZIELINSKI
Keyword(s):  
Equivalence Relation ◽  
Isometry Group ◽  
Alternative Proof ◽  
Orbit Equivalence ◽  
Locally Compact ◽  
Continuous Action ◽  
Polish Groups ◽  
Infinite Dimensional ◽  
Open Question ◽  
New Criterion

Abstract It is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. We answer this question positively for the class of all Polish groups that embed in the isometry group of a locally compact metric space. This class contains all non-archimedean Polish groups, for which we provide an alternative proof based on a new criterion for non-essential countability. Finally, we provide the following variant of a theorem of Solecki: every infinite-dimensional Banach space has a continuous action whose orbit equivalence relation is Borel but not essentially countable.

scholarly journals Orbit equivalence rigidity of equicontinuous systems

2016 ◽  
Vol 94 (2) ◽  
pp. 545-556 ◽  
Author(s):  
María Isabel Cortez ◽  
Konstantin Medynets
Keyword(s):  
Orbit Equivalence

Orbit equivalence and actions of

2006 ◽  
Vol 71 (1) ◽  
pp. 265-282 ◽  
Author(s):  
Asge Törnquist
Keyword(s):  
Probability Space ◽  
Free Groups ◽  
Orbit Equivalence ◽  
Borel Probability ◽  
Free Actions ◽  
Measure Preserving Transformations

AbstractIn this paper we show that there are “E0 many” orbit inequivalent free actions of the free groups , 2 ≤ n ≤ ∞ by measure preserving transformations on a standard Borel probability space. In particular, there are uncountably many such actions.

scholarly journals A rigidity result for crossed products of actions of Baumslag–Solitar groups

2015 ◽  
Vol 26 (14) ◽  
pp. 1550117
Author(s):  
Niels Meesschaert
Keyword(s):  
Probability Measure ◽  
Free Probability ◽  
Crossed Product ◽  
Crossed Products ◽  
Orbit Equivalence ◽  
Rigidity Result ◽  
Profinite Completions ◽  
Abelian Subgroups ◽  
Measure Equivalence ◽  
Measure Preserving Actions

Let [Formula: see text] and [Formula: see text] be two ergodic essentially free probability measure preserving actions of nonamenable Baumslag–Solitar groups whose canonical almost normal abelian subgroups act aperiodically. We prove that an isomorphism between the corresponding crossed product II1 factors forces [Formula: see text] when [Formula: see text] and [Formula: see text] when [Formula: see text]. This improves an orbit equivalence rigidity result obtained by Houdayer and Raum in [Baumslag–Solitar groups, relative profinite completions and measure equivalence rigidity, J. Topol. 8 (2015) 295–313].

scholarly journals Eigenvalues and strong orbit equivalence

2015 ◽  
Vol 36 (8) ◽  
pp. 2419-2440 ◽  
Author(s):  
MARÍA ISABEL CORTEZ ◽  
FABIEN DURAND ◽  
SAMUEL PETITE
Keyword(s):  
Equivalence Class ◽  
Quotient Group ◽  
Additive Group ◽  
Orbit Equivalence ◽  
Dimension Group ◽  
Torsion Free

We give conditions on the subgroups of the circle to be realized as the subgroups of eigenvalues of minimal Cantor systems belonging to a determined strong orbit equivalence class. Actually, the additive group of continuous eigenvalues $E(X,T)$ of the minimal Cantor system $(X,T)$ is a subgroup of the intersection $I(X,T)$ of all the images of the dimension group by its traces. We show, whenever the infinitesimal subgroup of the dimension group associated with $(X,T)$ is trivial, the quotient group $I(X,T)/E(X,T)$ is torsion free. We give examples with non-trivial infinitesimal subgroups where this property fails. We also provide some realization results.

On Constructing Ergodic Hyperfinite Equivalence Relations of Non-Product Type

2013 ◽  
Vol 56 (1) ◽  
pp. 136-147
Author(s):  
Radu-Bogdan Munteanu
Keyword(s):  
Equivalence Relation ◽  
Product Type ◽  
Bratteli Diagram ◽  
Equivalence Relations ◽  
Orbit Equivalence ◽  
Property A

AbstractProduct type equivalence relations are hyperfinitemeasured equivalence relations, which, up to orbit equivalence, are generated by product type odometer actions. We give a concrete example of a hyperfinite equivalence relation of non-product type, which is the tail equivalence on a Bratteli diagram. In order to show that the equivalence relation constructed is not of product type we will use a criterion called property A. This property, introduced by Krieger for non-singular transformations, is defined directly for hyperfinite equivalence relations in this paper.

scholarly journals Non-standard limits of graphs and some orbit equivalence invariants

2021 ◽  
Vol 4 ◽  
pp. 1235-1293
Author(s):  
Alessandro Carderi ◽  
Damien Gaboriau ◽  
Mikael de la Salle
Keyword(s):  
Orbit Equivalence
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