Full groups of Cantor minimal systems

Giordano, Thierry; Putnam, Ian F.; Skau, Christian

doi:10.1007/bf02810689

Cited by 125 publications

(199 citation statements)

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“…In fact, our dense subgroup comes directly from an example of Matui [20]. We recall some concepts and definitions from [10]. Let φ be an aperiodic (with all of its orbits infinite) homeomorphism of the Cantor space X .…”

Section: Automatic Continuitymentioning

confidence: 99%

“…In this subsection, E will always be ergodic and hyperfinite. By Dye's theorem, there exists only one hyperfinite, ergodic equivalence relation (up to isomorphism), so we have the flexibility to consider any measure-preserving, ergodic automorphism as the generator of E. In order to produce a finitely generated dense subgroup of [E], we use a specific minimal homeomorphism of the Cantor space as a topological model and the theory of topological full groups of minimal homeomorphisms, as developed by Giordano, Putnam and Skau [10]. In fact, our dense subgroup comes directly from an example of Matui [20].…”

Section: Automatic Continuitymentioning

confidence: 99%

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Topological properties of full groups

Kittrell

Tsankov

2009

Ergod. Th. Dynam. Sys.

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Abstract. We study full groups of countable, measure-preserving equivalence relations. Our main results include that they are all homeomorphic to the separable Hilbert space and that every homomorphism from an ergodic full group to a separable group is continuous. We also find bounds for the minimal number of topological generators (elements generating a dense subgroup) of full groups allowing us to distinguish between full groups of equivalence relations generated by free, ergodic actions of the free groups F m and F n if m and n are sufficiently far apart. We also show that an ergodic equivalence relation is generated by an action of a finitely generated group if an only if its full group is topologically finitely generated.

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Section: Automatic Continuitymentioning

confidence: 99%

Section: Automatic Continuitymentioning

confidence: 99%

Topological properties of full groups

Kittrell

Tsankov

2009

Ergod. Th. Dynam. Sys.

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“…In [GPS2], it was proved that [α] is a complete invariant for orbit equivalence and that [[α]] is a complete invariant for flip conjugacy.…”

Section: Definition 24 ([Gps2]mentioning

confidence: 99%

“…Moreover, it will be also pointed out that two systems are strong orbit equivalent if and only if the closures of the topological full groups are isomorphic. This is an approximate analogue of [GPS2,Corollary 4.4], in which it was proved that two systems are flip conjugate if and only if the topological full groups are isomorphic. In [GW], Glasner and Weiss proved that two Cantor minimal systems (X, α) and (Y, β) are weakly orbit equivalent if and only if the associated K 0 -groups are weakly isomorphic modulo infinitesimal subgroups.…”

Section: §1 Introductionmentioning

confidence: 97%

Approximate Conjugacy and Full Groups of Cantor Minimal Systems

Matui¹

2005

Publ. Res. Inst. Math. Sci.

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H. Lin and the author introduced the notion of approximate conjugacy of dynamical systems. In this paper, we will discuss the relationship between approximate conjugacy and full groups of Cantor minimal systems. An analogue of Glasner-Weiss's theorem will be shown. Approximate conjugacy of dynamical systems on the product of the Cantor set and the circle will also be studied. §1. IntroductionIn [LM1], several versions of approximate conjugacy were introduced for minimal dynamical systems on compact metrizable spaces. In this paper, we will restrict our attention on dynamical systems on zero or one dimensional compact spaces and discuss approximate conjugacy.Let X be the Cantor set. A homeomorphism α ∈ Homeo(X) is said to be minimal when it has no nontrivial closed invariant sets. We call (X, α) a Cantor minimal system. Giordano, Putnam and Skau introduced the notion of strong orbit equivalence for Cantor minimal systems in [GPS1], and showed that two systems are strong orbit equivalent if and only if the associated K 0 -groups are isomorphic. We will show that this theorem can be regarded as an approximate version of Boyle-Tomiyama's theorem ([GPS1, Theorem 2.4] or [BT, Theorem 3.2]). Moreover, it will be also pointed out that two systems

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“…Then, the Hopf-equivalence was found to play fundamental and crucial roles in analyses of measure-theoretical orbit structures of non-singular and bi-measurable transformations on Lebesgue spaces. And after that, it was discov ered by T. Giordano, R. Herman, I. Putnam and C. Skau [3,4,7] and E. Glasner and B. Weiss [5] that some relations similar to the Hopf-equivalence played fun damental and crucial roles even in the analyses of topological orbit structures of and then we define Aj = to 1 {t j } for 1 < j < k, and Bm = t~ 1 { sm } for 1 < m < 1.…”

mentioning

confidence: 99%