For an arbitrary countable discrete infinite group G, non-singular rank-one actions are introduced. It is shown that the class of non-singular rank-one actions coincides with the class of non-singular $(C,F)$ -actions. Given a decreasing sequence of cofinite subgroups in G with $\bigcap _{n=1}<^>\infty \bigcap _{g\in G}g\Gamma _ng<^>{-1}=\{1_G\}$ , the projective limit of the homogeneous G-spaces $G/\Gamma _n$ as $n\to \infty $ is a G-space. Endowing this G-space with an ergodic non-singular non-atomic measure, we obtain a dynamical system which is called a non-singular odometer. Necessary and sufficient conditions are found for a rank-one non-singular G-action to have a finite factor and a non-singular odometer factor in terms of the underlying $(C,F)$ -parameters. Similar conditions are also found for a rank-one non-singular G-action to be isomorphic to an odometer. Minimal Radon uniquely ergodic locally compact Cantor models are constructed for the non-singular rank-one extensions of odometers. Several concrete examples are constructed and several facts are proved that illustrate a sharp difference of the non-singular non-commutative case from the classical finite measure preserving one: odometer actions which are not of rank-one and factors of rank-one systems which are not of rank one; however, each probability preserving odometer is a factor of an infinite measure preserving rank-one system, etc.

This survey is a 2022 update of the 2008 version, with recent developments and new references.

It is shown that each non-compact locally compact second countable non-(T) group G possesses non-strongly ergodic weakly mixing IDPFT Poisson actions of arbitrary Krieger's type. These actions are amenable if and only if G is amenable. If G has the Haagerup property then (and only then) these actions can be chosen of 0-type. If G is amenable and unimodular then G has weakly mixing Bernoulli actions of any possible Krieger's type.

AbstractIt is shown that for a dense $G_\delta $ -subset of the subgroup of non-singular transformations (of a standard infinite $\sigma $ -finite measure space) whose Poisson suspensions are non-singular, the corresponding Poisson suspensions are ergodic and of Krieger’s type III1.

The classical Poisson functor associates to every infinite measure preserving dynamical system (X, μ, T) a probability preserving dynamical system (X*, μ*, T*) called the Poisson suspension of T. In this paper we generalize this construction: a subgroup Aut2(X, μ) of μ-nonsingular transformations T of X is specified as the largest subgroup for which T* is μ*-nonsingular. The topological structure of this subgroup is studied. We show that a generic element in Aut2(X, μ) is ergodic and of Krieger type III1. Let G be a locally compact Polish group and let A: G → Aut2(X, μ) be a G-action. We investigate dynamical properties of the Poisson suspension A* of A in terms of an affine representation of G associated naturally with A. It is shown that G has property (T), if and only if each nonsingular Poisson G-action admits an absolutely continuous invariant probability. If G does not have property (T), then for each generating probability κ on G and t > 0, a nonsingular Poisson G-action is constructed whose Furstenberg κ-entropy is t.

Let theta is an element of (0, 1) be an irrational number and let lambda := e2 pi i theta. For each well approximable irrational theta, we provide an explicit rank-1 construction of the lambda-rotation R lambda on the circle T. This solves "almost surely" a problem by del Junco. For every irrational theta, we construct explicitly a rank-1 transformation with an eigenvalue lambda. For every irrational theta, two infinite sigma-finite invariant measures mu lambda and mu 'lambda on T are constructed explicitly such that (T, mu lambda, R lambda) is rigid and of rank 1 and (T, mu 'lambda, R lambda) is of zero type and of rank 1. The centralizer of the latter system consists of just the powers of R lambda. Some versions of the aforementioned results are proved under an extra condition on boundedness of the sequence of cuts in the rank-1 construction.

It is shown that a locally compact second countable group G has the Haagerup property if and only if there exists a sharply weak mixing 0-type measure preserving free G-action T = (Tg)g∈G on an infinite σ-finite standard measure space (X, μ) admitting a T -Følner sequence (i.e. a sequence (An)∞n=1 of measured subsets of finite measure such that A1 ⊂ A2 ⊂ · · · , ⋃ ∞ n=1 An = X and limn→∞ supg∈K μ(TgAn△An) μ(An) = 0 for each compact K ⊂ G). A pair of groups H ⊂ G has property (T) if and only if there is a μ-preserving G-action S on X admitting an S-Følner sequence and such that S ↾ H is weakly mixing. These refine some recent results by Delabie-Jolissaint-Zumbrunnen and Jolissaint. 0. Introduction Throughout this paper G is a non-compact locally compact second countable group. It has the Haagerup property if there is a weakly continuous unitary representation V of G in a separable Hilbert space H such that limg→∞ V (g) = 0 in the weak operator topology and (∗) for each ǫ > 0 and every compact subset K ⊂ G, there is a unit vector ξ ∈ H such that supg∈G ‖V (g)ξ − ξ‖ < ǫ. Of course, the amenable groups have the Haagerup property. The class of discrete countable Haagerup groups contains the free groups and is closed under free products and wreath products [CoStVa]. For more information about the Haagerup property we refer to [Ch–Va]. There is a purely dynamical description of this property: G is Haagerup if and only if there exists a mixing non-strongly ergodic probability preserving free G-action [Ch–Va, Theorem 2.2.2] (see §1 for the definitions). Recently, an infinite measure preserving counterpart of this result was discovered in [DeJoZu]: Theorem A. G has the Haagerup property if and only if there is a 0-type measure preserving G-action T = (Tg)g∈G on an infinite σ-finite measure space (X,B, μ) admitting a sequence of non-negative unit vectors (ξn) ∞ n=1 in L (X,μ) such that limn→∞ supg∈K〈ξn ◦ Tg, ξn〉 = 1 for each compact K ⊂ G. We recall that T is called of 0-type if limg→∞ μ(TgA ∩ B) = 0 for all subsets A,B ∈ B of finite measure. In this paper we provide a much shorter alternative proof of Theorem A which is grounded on the Moore-Hill concept of restricted infinite products of probability measures [Hi]. We note that the 0-type for infinite measure preserving systems is a natural counterpart of the mixing for probability preserving systems. However unlike mixing, Typeset by AMS-TEX 1 the 0-type is not a “strong” asymptotic property. It implies neither weak mixing nor ergodicity. Moreover, the totally dissipative actions are all of 0-type. In view of that the description in Theorem A does not look sharp from the ergodic theory point of view. Our first main result in this work is the following finer ergodic criterion of the Haagerup property. Theorem B. The following are equivalent. (i) G has the Haagerup property. (ii) There exists a sharply weak mixing (conservative) 0-type measure preserving free G-action T on an infinite σ-finite standard measure space admitting an exhausting T -Følner sequence of subsets. (iii) There exists a a sharply weak mixing (conservative) 0-type measure preserving free G-action T on an infinite σ-finite standard measure space (X,B, μ) admitting a T -Følner sequence (An) ∞ n=1 such that μ(An) = 1 for all n ∈ N. We say that (An) ∞ n=1 is T -Følner if μ(An) <∞ and supg∈K μ(An△TgAn) μ(An) → 0 as n → ∞ for each compact subset K ⊂ G. If A1 ⊂ A2 ⊂ · · · and ⋃∞ n=1An = X , we say that (An) ∞ n=1 is exhausting. We note that sharp mixing (see §1 for the definition) implies ergodicity and weak mixing. To prove (the non-trivial part of) Theorem B, we apply the Moore-Hill construction [Hi] to the mixing non-strongly ergodic G-action from [Ch–Va, Theorem 2.2.2] (cf. the construction of II∞ ergodic Poisson suspensions of countable amenable groups from [DaKo]). Then we observe that the action T that we obtain is IDPFT (see §1 and [DaLe], where such actions were introduced). Hence, by the properties of IDPFT systems, T is sharply weak mixing whenever we show that it is conservative. To show the conservativeness of T is remains to choose the parameters of the Moore-Hill construction in an a appropriate way. As a corollary from Theorem B, we obtain one more dynamical characterization of the Haagerup property in terms of Poisson actions. Corollary C. G has the Haagerup property if and only if there exists a mixing (probability preserving) Poisson G-action that is not strongly ergodic. Our next purpose is to obtain a “parallel” characterization of property (T) which is a reciprocal to the Haagerup property. We recall [Jo1, Definition 1.1] that given a non-compact closed subgroup H of G, the pair H ⊂ G has property (T) if for each unitary representation V of G satisfying (∗), there is a unit vector which is invariant under V (h) for every h ∈ H . Using the techniques developed for proving Theorem B we obtain an ergodic (non-spectral) characterization of Kazhdan pairs that refines a spectral characterization from [Jo2]. Theorem D. (i) If a pair H ⊂ G has property (T) then each measure preserving G-action S = (Sg)g∈G on a σ-finite infinite standard measure space (Y,C, ν), such that S ↾ H := (Sh)h∈H has no invariant subsets of positive finite measure, admits no S-Følner sequences. 1Conservativeness, ergodicity, weak mixing and sharp weak mixing are not spectral invariants of the underlying dynamical systems. Hence the principal difference of Theorem B from Theorem A is that it provides non-spectral ergodic characterization of the Haagerup property. 2 (ii) If a pair H ⊂ G does not have property (T) then there is a measure preserving G-action S on a σ-finite infinite measure space which has an exhausting S-Følner sequence and such that S ↾ H is weakly mixing. Let us say that S ↾ H is of weak 0-type if there is a subsequence hn → ∞ in H such that limn→∞ ν(ShnA ∩ B) = 0 for all subsets A,B ∈ C of finite measure. Then replacing the “has no invariant subsets of positive finite measure” in (i) with a stronger “is of weak 0-type”, and the “weakly mixing” in (ii) with a weaker “of weak 0-type” we obtain exactly [Jo2, Theorem 1.5]. Corollary E. A pair H ⊂ G has property (T) if and only if every (probability preserving) Poisson G-action with weakly mixing H-subaction is strongly ergodic. The same is also true with “ergodic” in place of “weakly mixing”. The outline of the paper is as follows. In Section 1 we state all necessary definitions related to the basic dynamical concepts of group actions both in the nonsingular and and finite measure preserving cases, restricted infinite powers of probability measures, IDPFT actions and Poisson actions. In Section 2 we prove Theorems B and Corollary C. Section 3 is devoted to the proof of Theorems D and Corollary E. 1. Definitions and preliminaries Nonsingular and measure preserving G-actions. Nonsingular actions appear in the proof of Theorem B. We remind several basic concepts related to them. Definition 1.1. Let S = (Sg)g∈G be a nonsingular G-action on a standard probability space (Z,F, κ). (i) S is called totally dissipative if the partition of Z into the S-orbits is measurable and the S-stabilizer of a.e. point is compact, i.e. there is a measurable subset of Z which meets a.e. S-orbit exactly once, and for a.e. z ∈ Z, the subgroup {g ∈ G | Sgz = z} is compact in G. (ii) S is called conservative if there is no any S-invariant subset A ⊂ Z of positive measure such that the restriction of S to A is totally dissipative. (iii) There is a unique (mod 0) partition of X into two invariant subsets D(S) and C(S) such that S ↾ D(S) is totally dissipative and S ↾ D(S) is conservative. We call D(S) and C(S) the dissipative and conservative part of S respectively. (iv) S is called ergodic if each measurable S-invariant subset of Z is either μ-null or μ-conull. (v) S is called weakly mixing if for each ergodic probability preserving G-action R = (Rg)g∈G, the product G-action (Sg ×Rg)g∈G is ergodic. (vi) S is called properly ergodic if it is ergodic and κ is not concentrated on a single orbit. (vii) S is called sharply weak mixing [DaLe] if it is properly ergodic and for each ergodic conservative nonsingular G-action R = (Rg)g∈G on a nonatomic probability space, the product G-action (Sg × Rg)g∈G is either ergodic or totally dissipative. We also remind some concepts related to finite measure preserving actions. Definition 1.2. Suppose that κ(Z) = 1 and κ ◦ Sg = κ for all g ∈ G. (i) S is called mixing if limg→∞ κ(SgA ∩B) = μ(A)μ(B) for all A,B ∈ F. 3 (ii) A sequence of Borel subsets (An) ∞ n=1 in X of strictly positive measure is called T -asymptotically invariant if for each compact subset K ⊂ G, we have that supg∈K κ(An△TgAn) → 0 as n→ ∞. (iii) T is called strongly ergodic if each T -asymptotically invariant sequence (An) ∞ n=1 is trivial, i.e. limn→∞ κ(An)(1− μ(An)) = 0. We now state a corollary from the Schmidt-Walters theorem [ScWa, Theorem 2.3]. Lemma 1.3. Let S = (Sg)g∈G be a mixing measure preserving action on a standard probability space (Y,C, ν). Then for each ergodic non-totally dissipative nonsingular G-action R = (Rg)g∈G, the product G-action S ×R := (Sg ×Rg)g∈G is ergodic. Proof. We first note that a mixing action is properly ergodic. Hence if R = (Rg)g∈G is properly ergodic then the claim of the proposition follows immediately from [ScWa, Theorem 2.3]. If R is not properly ergodic then there is a noncompact subgroup H in G such that R is isomorphic to the G-action by left translations on the coset space G/H endowed with a Haar measure. Hence S ×R is ergodic if and only if the H-action (S(h))h∈H on (Y,C, ν) is ergodic. The later holds because S is mixing. Corollary 1.4. Let S = (Sg)g∈G be a mixing measure preserving action on a standard probability space (Y,C, ν) and let R = (Rg)g∈G be a nonsingular G-action on a standard probability space (Z,D, κ). The following holds. (i) D(S ×R) = Y ×D(R) and C(S ×R) = Y × C(R). (ii) If R is conservative and F : Y × Z → C is an (S × R)-in

AbstractIt is proved that each Gaussian cocycle over a mildly mixing Gaussian transformation is either a Gaussian coboundary or sharply weak mixing. The class of non-singular infinite direct productsTof transformations$T_n$,$n\in \mathbb N$, of finite type is studied. It is shown that if$T_n$is mildly mixing,$n\in \mathbb N$, the sequence of Radon–Nikodym derivatives of$T_n$is asymptotically translation quasi-invariant andTis conservative then the Maharam extension ofTis sharply weak mixing. This technique provides a new approach to the non-singular Gaussian transformations studied recently by Arano, Isono and Marrakchi.

Given an infinite countable discrete amenable group $\Gamma$, we construct explicitly sharply weak mixing nonsingular Poisson $\Gamma$-actions of each Krieger's type: $III_\lambda$, for $\lambda\in[0,1]$, and $II_\infty$. The result is new even for $\Gamma=\Bbb Z$. As these Poisson suspension actions are over very special dissipative base, we obtain also new examples of sharply weak mixing nonsingular Bernoulli $\Gamma$-actions and IDPFT systems of each possible Krieger's type.

Let E be a subset of positive integers such that E∩{1, 2} 6= ∅. A weakly mixing finite measure preserving flow T = (Tt)t∈R is constructed such that the set of spectral multiplicities (of the corresponding Koopman unitary representation generated by T ) is E. Moreover, for each non-zero t ∈ R, the set of spectral multiplicities of the transformation Tt is also E. These results are partly extended to actions of some other locally compact second countable Abelian groups.

For each {\it well approximable} irrational $\theta$, we provide an explicit rank-one construction of the $e^{2\pi i\theta}$-rotation $R_\theta$ on the circle $\Bbb T$. This solves "almost surely" a problem by del Junco. For {\it every} irrational $\theta$, we construct explicitly a rank-one transformation with an eigenvalue $e^{2\pi i\theta}$. For every irrational $\theta$, two infinite $\sigma$-finite invariant measures $\mu_\theta$ and $\mu_{\theta}'$ on $\Bbb T$ are constructed explicitly such that $(\Bbb T,\mu_\theta, R_\theta)$ is {\it rigid} and of rank one and $(\Bbb T,\mu_\theta', R_\theta)$ is of {\it zero type} and of rank one. The centralizer of the latter system consists of just the powers of $R_\theta$. Some versions of the aforementioned results are proved under an extra condition on boundedness of the sequence of cuts in the rank-one construction.

Let Γ be an amenable countable discrete group. Fix an ergodic free non-singular action of Γ on a nonatomic standard probability space. Let G be a compactly generated locally compact second countable group such that the closure of the group of inner automorphisms of G is compact in the natural topology. It is shown that there exists a bounded ergodic G-valued cocycle of Γ.

For each {\it well approximable} irrational $\theta$, we provide an explicit rank-one construction of the $e^{2\pi i\theta}$-rotation $R_\theta$ on the circle $\Bbb T$. This solves "almost surely" a problem by del Junco. For {\it every} irrational $\theta$, we construct explicitly a rank-one transformation with an eigenvalue $e^{2\pi i\theta}$. For every irrational $\theta$, two infinite $\sigma$-finite invariant measures $\mu_\theta$ and $\mu_{\theta}'$ on $\Bbb T$ are constructed explicitly such that $(\Bbb T,\mu_\theta, R_\theta)$ is {\it rigid} and of rank one and $(\Bbb T,\mu_\theta', R_\theta)$ is of {\it zero type} and of rank one. The centralizer of the latter system consists of just the powers of $R_\theta$. Some versions of the aforementioned results are proved under an extra condition on boundedness of the sequence of cuts in the rank-one construction.

It is shown that for a dense $G_\delta$-subset of the subgroup of nonsingular transformations (of a standard infinite $\sigma$-finite measure space) whose Poisson suspensions are nonsingular, the corresponding Poisson suspensions are ergodic and of Krieger's type $III_1$.

It is shown that each conservative nonsingular Bernoulli shift is either of type II_1 or III_1. Moreover, in the latter case the corresponding Maharam extension of the shift is a K-automorphism. This extends earlier results obtained by Z. Kosloff for the equilibrial shifts. Nonequilibrial shifts of type III_1 are constructed. We further generalize (partly) the main results to nonsingular Markov shifts.

Let G be a discrete countable infinite group. We show that each topological (C, F)-action T of G on a locally compact non-compact Cantor set is a free minimal amenable action admitting a unique up to scaling non-zero invariant Radon measure (answer to a question by Kellerhals, Monod and Rørdam). We find necessary and sufficient conditions under which two such actions are topologically conjugate in terms of the underlying (C, F)-parameters. If G is linearly ordered abelian, then the topological centralizer of T is trivial. If G is monotileable and amenable, denote by $${{\cal A}_G}$$ the set of all probability preserving actions of G on the unit interval with Lebesgue measure and endow it with the natural topology. We show that the set of (C, F)-parameters of all (C, F)-actions of G furnished with a suitable topology is a model for $${{\cal A}_G}$$ in the sense of Foreman, Rudolph and Weiss. If T is a rank-one transformation with bounded sequences of cuts and spacer maps, then we found simple necessary and sufficient conditions on the related (C, F)-parameters under which (i) T is rigid, (ii) T is totally ergodic. An alternative proof is found of Ryzhikov’s theorem that if T is totally ergodic and a non-rigid rank-one map with bounded parameters, then T has MSJ. We also give a more general version of the criterion (by Gao and Hill) for isomorphism and disjointness of two commensurate non-rigid totally ergodic rank-one maps with bounded parameters. It is shown that the rank-one transformations with bounded parameters and no spacers over the last subtowers is a proper subclass of the rank-one transformations with bounded parameters.

Let $G$ be an amenable discrete countable infinite group, $A$ a finite set, and $(\mu_g)_{g\in G}$ a family of probability measures on $A$ such that $\inf_{g\in G}\min_{a\in A}\mu_g(a)>0$. It is shown (among other results) that if the Bernoulli shiftwise action of $G$ on the infinite product space $\bigotimes_{g\in G}(A,\mu_g)$ is nonsingular and conservative then it is weakly mixing. This answers in positive a question by Z.~Kosloff who proved recently that the conservative Bernoulli $\Bbb Z^d$-actions are ergodic. As a byproduct, we prove a weak version of the pointwise ratio ergodic theorem for nonsingular actions of $G$.

We introduce concepts of Radon MSJ and Radon disjointness for infinite Radon measure preserving homeomorphisms of the locally compact Cantor space. We construct an uncountable family of pairwise Radon disjoint infinite Chacon like transformations. Every such transformation is Radon strictly ergodic, totally ergodic, asymmetric (not isomorphic to its inverse), has Radon MSJ and possesses Radon joinings whose ergodic components are not joinings.

Let$\unicode[STIX]{x1D6E4}$be a lattice in a simply connected nilpotent Lie group$G$. Given an infinite measure-preserving action$T$of$\unicode[STIX]{x1D6E4}$and a ‘direction’ in$G$(i.e. an element$\unicode[STIX]{x1D703}$of the projective space$P(\mathfrak{g})$of the Lie algebra$\mathfrak{g}$of$G$), some notions of recurrence and rigidity for$T$along$\unicode[STIX]{x1D703}$are introduced. It is shown that the set of recurrent directions${\mathcal{R}}(T)$and the set of rigid directions for$T$are both$G_{\unicode[STIX]{x1D6FF}}$. In the case where$G=\mathbb{R}^{d}$and$\unicode[STIX]{x1D6E4}=\mathbb{Z}^{d}$, we prove that (a) for each$G_{\unicode[STIX]{x1D6FF}}$-subset$\unicode[STIX]{x1D6E5}$of$P(\mathfrak{g})$and a countable subset$D\subset \unicode[STIX]{x1D6E5}$, there is a rank-one action$T$such that$D\subset {\mathcal{R}}(T)\subset \unicode[STIX]{x1D6E5}$and (b)${\mathcal{R}}(T)=P(\mathfrak{g})$for a generic infinite measure-preserving action$T$of$\unicode[STIX]{x1D6E4}$. This partly answers a question from a recent paper by Johnson and Şahin. Some applications to the directional entropy of Poisson actions are discussed. In the case where$G$is the Heisenberg group$H_{3}(\mathbb{R})$and$\unicode[STIX]{x1D6E4}=H_{3}(\mathbb{Z})$, a rank-one$\unicode[STIX]{x1D6E4}$-action$T$is constructed for which${\mathcal{R}}(T)$is not invariant under the natural ‘adjoint’$G$-action.