A Brief Introduction to Amenable Equivalence Relations

The notion of an amenable equivalence relation was introduced by Zimmer in the course of his analysis of orbit equivalence relations in ergodic theory (see [12]). Recently it played an important role in Monod’s striking family of examples of nonamenable groups which do not contain nonabelian free subgroups. If A is a subring of R, define H(A) to be the group of all piecewise PSL2(A) homeomorphisms of the real projective line which fix the point at infinity. Theorem 1.1. [17] If A is any dense subring of R, then H(A) is nonamenable. Moreover, if f, g ∈ H(R), then either 〈f, g〉 is metabelian or else contains an infinite rank free abelian subgroup. In particular, H(R) does not contain a nonabelian free subgroup.