Geodesic
Amenable actions of nonamenable groups
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XIII, Tome 326 (2005), pp. 85-96

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We give two ways of constructing amenable (in the sense of Greenleaf) actions of nonamenable groups. In the first part of the paper we construct a class of faithful transitive amenable actions of the free group using Schreier graphs. In the second part we show that every finitely generated residually finite group can be embedded into a bigger residually finite group, which acts level-transitively on a locally finite rooted tree, so that the induced action on the boundary of the tree is amenable on every orbit. 
@article{ZNSL_2005_326_a6,
     author = {R. I. Grigorchuk and V. V. Nekrashevych},
     title = {Amenable actions of nonamenable groups},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {85--96},
     publisher = {mathdoc},
     volume = {326},
     year = {2005},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2005_326_a6/}
}
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R. I. Grigorchuk; V. V. Nekrashevych. Amenable actions of nonamenable groups. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XIII, Tome 326 (2005), pp. 85-96. http://geodesic.mathdoc.fr/item/ZNSL_2005_326_a6/