On a measurable analogue of small topological full groups

Maître, François Le

doi:10.1016/j.aim.2018.05.008

Cited by 4 publications

(2 citation statements)

References 36 publications

(43 reference statements)

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“…The approach of looking at orientations of orbits was suggested to me by F. Le Maître, who was in turn inspired by work by Belinskaya [1], both working in the measure-theoretic setting. Several of the results in this paper mimic those of [10, Section 4.3] in particular. The novelty in the topological dynamics setting is to establish that various sets of interest are actually clopen.…”

Section: Introductionsupporting

confidence: 80%

Orientation of Piecewise Powers of a Minimal Homeomorphism

Reid

2021

J. Aust. Math. Soc.

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We show that, given a compact minimal system $(X,g)$ and an element h of the topological full group $\tau [g]$ of g, the infinite orbits of h admit a locally constant orientation with respect to the orbits of g. We use this to obtain a clopen partition of $(X,G)$ into minimal and periodic parts, where G is any virtually polycyclic subgroup of $\tau [g]$ . We also use the orientation of orbits to give a refinement of the index map and to describe the role in $\tau [g]$ of the submonoid generated by the induced transformations of g. Finally, we consider the problem, given a homeomorphism h of the Cantor space X, of determining whether or not there exists a minimal homeomorphism g of X such that $h \in \tau [g]$ .

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Section: Introductionsupporting

confidence: 80%

Orientation of Piecewise Powers of a Minimal Homeomorphism

Reid

2021

J. Aust. Math. Soc.

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“…However thanks to the work of Fremlin [Fre02, 384D], Dye's reconstruction theorem is now known to hold for a much wider class of groups of measure-preserving transformations, e.g. the so-called L 1 full groups considered in [LM18]. It would be interesting to know if Theorem A can be extended to such groups, possibly at the cost of requiring continuity of the boolean action.…”

mentioning

confidence: 99%

More Polish full groups

Carderi¹,

Maître²

2016

Topology and its Applications

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We associate to every action of a Polish group on a standard probability space a Polish group that we call the orbit full group. For discrete groups, we recover the wellknown full groups of pmp equivalence relations equipped with the uniform topology. However, there are many new examples, such as orbit full groups associated to measure preserving actions of locally compact groups. In fact, we show that such full groups are complete invariants of orbit equivalence.We give various characterizations of the existence of a dense conjugacy class for orbit full groups, and we show that the ergodic ones actually have a unique Polish group topology. Furthermore, we characterize ergodic full groups of countable pmp equivalence relations as those admitting non-trivial continuous character representations.

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Quantitative measure equivalence between amenable groups

Delabie¹,

Koivisto²,

Maître³

et al. 2022

Annales Henri Lebesgue

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We initiate a quantitative study of measure equivalence (and orbit equivalence) between finitely generated groups, which extends the classical setting of L p measure equivalence. In this paper, our main focus will be on amenable groups, for which we prove both rigidity and flexibility results.On the rigidity side, we prove a general monotonicity property satisfied by the isoperimetric profile, which implies in particular its invariance under L 1 measure equivalence. This yields explicit "lower bounds" on how integrable a measure coupling between two amenable groups can be. This result also has an unexpected application to geometric group theory: the isoperimetric profile turns out to be monotonous under coarse embedding between amenable groups. This

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On a measurable analogue of small topological full groups

Cited by 4 publications

References 36 publications

Orientation of Piecewise Powers of a Minimal Homeomorphism

Orientation of Piecewise Powers of a Minimal Homeomorphism

More Polish full groups

Quantitative measure equivalence between amenable groups

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