Thompson's Group <i>F</i> is Not Liouville

Kaimanovich, Vadim A.

doi:10.1017/9781316576571.013

Cited by 7 publications

(3 citation statements)

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“…We denote by F the Thompson group F . In [5] and [8], authors show that the Schreier graph of the action of F on the orbit of 1/2 is not Liouville with respect to any measure of finite support. Here we show that there are measures with infinite support that make this action Liouville.…”

Section: Applications To the Thompson Group Fmentioning

confidence: 99%

“…While many of the definitions are equivalent for Cayley graphs, they are not equivalent when we pass to actions. The main reason of our current definition is a recent approach to amenability developed by Kaimanovich in [5]. In order to show that a group is not amenable it is sufficient to find an action which does not admit any non-degenerate Liouville measure.…”

Section: Introductionmentioning

confidence: 99%

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Infinitely supported Liouville measures of Schreier graphs

Juschenko

Zheng

2018

Groups Geom. Dyn.

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We provide equivalent conditions for Liouville property of actions of groups. As an application, we show that there is a Liouville measure for the action of the Thompson group F on dyadic rationals. This result should be compared with a recent result of Kaimanovich, where he shows that the action of the Thompson group F on dyadic rationals is not Liouville for all finitely supported measures. As another application we show that there is a Liouville measure for lamplighter actions. This gives more examples of non-amenable Liouville actions.

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Section: Applications To the Thompson Group Fmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Infinitely supported Liouville measures of Schreier graphs

Juschenko

Zheng

2018

Groups Geom. Dyn.

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“…We would like to notice that the study of Schreier graphs of F has already provided valuable information about this group. For example, a description of a family of Schreier graphs of F by Dmytro Savchuk in [16] is used in study of Poisson-Furstenberg boundary of F , see [15], [11], [9] and recent preprints [17] and [10].…”

Section: Introductionmentioning

confidence: 99%