Geometriae Dedicata

, Volume 175, Issue 1, pp 355–372 | Cite as

Schreier graphs of actions of Thompson’s group \(F\) on the unit interval and on the Cantor set

Original Paper

Abstract

Schreier graphs of the actions of Thompson’s group \(F\) on the orbits of all points of the unit interval and of the Cantor set with respect to the standard generating set \(\{x_0,x_1\}\) are explicitly constructed. The closure of the space of pointed Schreier graphs of the action of \(F\) on the orbits of dyadic rational numbers and corresponding Schreier dynamical system are described. In particular, we answer the question of Grigorchuk on the Cantor–Bendixson rank of the underlying space of the Schreier dynamical system in the context of \(F\). As applications we prove that the pointed Schreier graphs of points from \((0,1)\) are amenable, have infinitely many ends, and are pairwise non-isomorphic. Moreover, we prove that points \(x,y\in (0,1)\) have isomorphic non-pointed Schreier graphs if and only if they belong to the same orbit of \(F\).

Keywords

Schreier graphs Thompson groups Schreier dynamical systems 

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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of South FloridaTampaUSA

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