Haagerup property and Kazhdan pairs via ergodic infinite measure preserving actions

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