Take the 2-minute tour ×  Question. Give a (possibly elementary)
example of a probability measure preserving action $\rho\colon G
\curvearrowright X$ of a finitely-generated discrete group $G$
on a standard borel space $X$ with a probability measure, such
that    1. the equivalence relation generated by $\rho$ is
ergodic and amenable,   2. the action $\rho$ is faithful,   3.
the group $G$ is non-amenable.  A friend of mine asked me this
question couple of days ago, which led us to another question,
but perhaps there is an easier way to provide an example.
share|improve this question This question is discussed in:
Alexander S. Kechris "Global Aspects of Ergodic Group Actions
and Equivalence Relations", pp. 31-34. It is available on line
here: citeseerx.ist.psu.edu/viewdoc/summary?doi= –  Jesse
Peterson Mar 27 '11 at 7:18 You can use the construction known
as Mackey range: take a cocycle of an amenable equivalence
relation with values in a non-amenable group G. Then G induces
an amenable action on the space of