We shall have to do with certain topology in the set of finitely generated groups which arises in the problems related to approximation properties of (transformation) groups. This topology gives us new insight concerning the amenable groups and their actions.

Our approach will be illustrated by proving a version of the one-tower Halmos-Kakutani-Rokhlin property for majority of the known amenable groups. Incidentally we shall get a clear (as I hope) understanding of

1. 1) the existence phenomenon of non-elementary amenable groups and

2. 2) the nature of such groups known up to now.

There are 3 sources and 3 constituents of the arguments leading to our construction of the one-tower Halmos-Kakutani-Rokhlin (HKR) property for some non-elementary amenable groups. These (both sources and constituents) are:

1. 1) the notion of local isomorphism (and corresponding topology) for finitely generated groups,

2. 2) Grigorchuk's construction of the finitely generated groups having intermediate growth,

3. 3) Caroline Series' proof of the HKR property for solvable groups.

GROUP ACTIONS WITH THE FREE APPROXIMATION PROPERTY AND LOCAL ISOMORPHISM OF GROUPS

A partition ξ of a measure space (X, µ) is (called) nonsingular if the saturation mapping corresponding to ξ transforms the class of measure zero sets into itself. An action T of a countable group G in (X, µ) is said to be approximable if its trajectory partition is the intersection (i.e. the greatest lower bound) of some decreasing sequence of measurable nonsingular partitions.