Groups of piecewise projective homeomorphisms

Monod, Nicolas

doi:10.1073/pnas.1218426110

Cited by 89 publications

(82 citation statements)

References 38 publications

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“…In this section we show that this is not the only obstruction. We consider a group H of piecewise projective orientation-preserving homeomorphisms of R intruduced in [Mon13]. A self-homeomorphisms f of R belongs to this group if there exist intervals I 1 , .…”

Section: Connection With Topological Full Groups and Liouville Propermentioning

confidence: 99%

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Extensive amenability and an application to interval exchanges

Juschenko¹,

Bon

Monod

2016

Ergod. Th. Dynam. Sys.

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Extensive amenability is a property of group actions which has recently been used as a tool to prove amenability of groups. We study this property and prove that it is preserved under a very general construction of semidirect products. As an application, we establish the amenability of all subgroups of the group IET of interval exchange transformations that have angular components of rational rank ≤ 2.In addition, we obtain a reformulation of extensive amenability in terms of inverted orbits and use it to present a purely probabilistic proof that recurrent actions are extensively amenable. Finally, we study the triviality of the Poisson boundary for random walks on IET and show that there are subgroups G < IET admitting no finitely supported measure with trivial boundary.

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Section: Connection With Topological Full Groups and Liouville Propermentioning

confidence: 99%

“…The fact that H R is not extensively amenable follows from the following Theorem because it was proved in [Mon13] that H is not amenable. The proof relies on the main result in [JNdlS13].…”

Section: Connection With Topological Full Groups and Liouville Propermentioning

confidence: 99%

Extensive amenability and an application to interval exchanges

Juschenko¹,

Bon

Monod

2016

Ergod. Th. Dynam. Sys.

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“…The following theorem of Monod is remarkable both for its strength and for the simplicity of the proof 302 Theorem 5.9.21 (Monod,[234]). For every ring A ≤ R containing 1, A ≠ Z, the group H(A) is non-amenable and contains no free non-cyclic subgroups.…”

Section: Two Church-rosser Presentations Of Fmentioning

confidence: 99%

Combinatorial Algebra: Syntax and Semantics

Sapir¹,

Guba²,

Volkov³

2014

Springer Monographs in Mathematics

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“…Monod constructs in [34] a class of groups of piecewise projective homeomorphisms HpAq (where A is a subring of R). By comparing the action of HpAq on the projective line P 1 pRq with that of P SL 2 pAq, he proves that it is non-amenable for A ‰ Z and without free subgroups for all A.…”

Section: Introductionmentioning

confidence: 99%

Non-triviality of the Poisson boundary of random walks on the group of Monod

Bogdan¹

2019

Ergod. Th. Dynam. Sys.

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We give sufficient conditions for the non-triviality of the Poisson boundary of random walks on HpZq and its subgroups. The group HpZq is the group of piecewise projective homeomorphisms over the integers defined by Monod. For a finitely generated subgroup H of HpZq, we prove that either H is solvable, or every measure on H with finite first moment that generates it as a semigroup has non-trivial Poisson boundary. In particular, we prove the non-triviality of the Poisson boundary of measures on Thompson's group F that generate it as a semigroup and have finite first moment, which answers a question by Kaimanovich.

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Groups of piecewise projective homeomorphisms

Cited by 89 publications

References 38 publications

Extensive amenability and an application to interval exchanges

Extensive amenability and an application to interval exchanges

Combinatorial Algebra: Syntax and Semantics

Non-triviality of the Poisson boundary of random walks on the group of Monod

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