FUNNY RANK-ONE WEAK MIXING FOR NONSINGULAR ABELIAN ACTIONS Alexandre I. Danilenko Abstract. We construct fanny rank-one infinite measure preserving free actions T of a countable Abelian group G satisfying each of the following properties: (1) T g 1 ×···×T g k is ergodic for each finite sequence g 1 ,...,g k of G-elements of infinite order, (2) T ×T is nonconservative, (3) T ×T is nonergodic but all k-fold Cartesian products are conservative, and the L ∞ -spectrum of T is trivial, (4) for each g of infinite order, all k-fold Cartesian products of T g are ergodic, but T 2g × T g is nonconservative. A topological version of this theorem holds. Moreover, given an AT-flow W , we construct nonsingular G-actions T with the similar properties and such that the associated flow of T is W . Orbit theory is used in an essential way here. 0. Introduction The goal of this work is to construct infinite measure preserving and non- singular fanny rank-one free actions of countable Abelian groups with various dynamical properties. The construction of these actions is based on a common idea: every one appears as an inductive limit of some partially defined actions associated to certain two sequences (C n ) and (F n ) of finite subsets in the group. We call them (C, F )-actions. It is worthwhile to remark that the (C, F )-actions appear as minimal topological actions on locally compact totally disconnected spaces. Moreover, they are uniquely ergodic, i.e. they admit a unique (up to scaling) invariant σ-finite Radon measure (Borel measure which is finite on the compact subsets). Now we record our main result about infinite measure preserving actions. Theorem 0.1. Let G be a countable Abelian group. Given i ∈{1,..., 5}, there exists a funny rank one infinite measure preserving free (C, F )-action T = {T g } g∈G of G such that the property (i) of the following list is satisfied: (1) for every g ∈ G of infinite order, the transformation T g has infinite er- godic index, i.e. all its k-fold Cartesian products are ergodic, 1991 Mathematics Subject Classification. Primary 28D15, 28D99. Key words and phrases. Weak mixing, AT-flow. The work was supported in part by INTAS 97-1843 and CRDF-grant UM1-2092. Typeset by A M S-T E X 1