Orbit equivalence superrigidity for type III 0 actions

. We prove the ﬁrst orbit equivalence superrigidity results for actions of type III λ when λ (cid:2)= 1. These actions arise as skew products of actions of dense subgroups of SL (n , R ) on the sphere S n − 1 and they can have any prescribed associated ﬂow.


Introduction and main results
An essentially free, ergodic, non-singular action G (X, μ) of a countable group G on a standard probability space is said to be orbit equivalence (OE) superrigid if the group G and its action on (X, μ) can be entirely retrieved from the orbit equivalence relation R(G X) = {(x, g · x) | x ∈ X, g ∈ G}. Especially in the case where μ is a G-invariant probability measure, several OE-superrigidity theorems were proven in the context of Popa's deformation/rigidity theory; see, for example, [DIP19,Ioa08,Pop05,Pop06].
Beyond the probability measure-preserving setting, OE-superrigidity results are more scarce; see, for example, [DV21,Ioa14,PV08]. In all these cases, the group action G (X, μ) has one of the following Krieger types: II 1 , II ∞ or III 1 (see §2 for definitions). There is a conceptual reason why it is harder to prove OE superrigidity for actions of type III λ , λ ∈ [0, 1). One typically proves OE superrigidity for G (X, μ) by showing that every measurable 1-cocycle ω : G × X → with values in an arbitrary countable group is cohomologous to a group homomorphism δ : G → . When the measure μ is not G-invariant, the logarithm of the Radon-Nikodym derivative d(g −1 · μ)/dμ provides a 1-cocycle ω : G × X → R. In type III λ with λ ∈ [0, 1), this 1-cocycle is 'essentially' similar to a 1-cocycle with values in a countable group. Therefore, cocycle superrigidity tends to fail.
In this paper, we obtain the first OE-superrigidity results in type III λ when λ = 1. In Theorem C, we prove OE superrigidity for the affine action of dense subgroups G < SL(n, R) R n on X = R n . These actions can be of type III λ for any λ ∈ (0, 1]. In this result, OE superrigidity holds in its strongest possible form: for every essentially free, 2 S. Vaes and B. Verjans ergodic, non-singular action (Z, ζ ) that is stably orbit equivalent to G (X, μ), there exists an injective group homomorphism δ : G → such that (Z, ζ ) is isomorphic to the induction of G (X, μ) along δ. In Theorem C, we also prove that this strongest form of OE superrigidity can basically never hold for actions of type III 0 . In type III 0 , it is necessary to further reduce the other action (Z, ζ ): after writing Z as an induction of 0 Z 0 , we need to take a quotient 0 / Z 0 / by a normal subgroup whose action on Z 0 admits a fundamental domain, before arriving at an action that is conjugate with G (X, μ). This leads to a second, slightly weaker version of OE superrigidity that we denote, without too much inspiration, as OE superrigidity (v2). In Theorem A and Corollary B, we then prove that natural skew product actions of dense subgroups of SL(n, R) are OE superrigid (v2), of type III 0 , with any prescribed associated flow.
Before stating our main results, we make this terminology more precise. Note that the concepts of (stable) orbit equivalence, induced actions, conjugate actions, etc., are recalled in §2.
Beyond the probability measure-preserving setting, one cannot distinguish between orbit equivalence and stable orbit equivalence. Therefore, induced actions will appear in any OE-superrigidity statement. We thus formally define the following property for a free, ergodic, non-singular action G (X, μ). • OE superrigidity (v1) of G (X, μ): any free, ergodic, non-singular action that is stably orbit equivalent to G (X, μ) is conjugate to an induction of G (X, μ). As we prove in Theorem C, this (v1) of OE superrigidity can basically never hold for actions of type III 0 , but does hold for several actions of type III λ with λ ∈ (0, 1]. For actions of type III 0 , unavoidably the following extra freedom is needed, accommodating the canonical stable orbit equivalences that come with induction and with quotients by normal subgroups whose action admits a fundamental domain. • OE superrigidity (v2) of G (X, μ): if a free, ergodic, non-singular action (Z, ζ ) is stably orbit equivalent to G (X, μ), there exist subgroups < 0 < and a non-negligible Z 0 ⊂ Z such that Z is induced from 0 Z 0 , ¡ 0 is normal, the action Z 0 admits a fundamental domain and G (X, μ) is conjugate with 0 / Z 0 / . To obtain technically less involved statements, one may restrict to simple actions: in Definition 4.1, we say that a free, ergodic, non-singular action G (X, μ) is simple if the action is not induced and if G has no non-trivial normal subgroups whose action on (X, μ) admits a fundamental domain. Then both versions of OE superrigidity for a simple action G (X, μ) immediately imply that any stably orbit equivalent simple action must be conjugate to G X, bringing us back to a statement that looks similar to the probability measure-preserving setting.
We use the following skew product construction to obtain OE-superrigid actions of type III 0 . Given any non-singular ergodic action G (X, μ) of type III 1 , with logarithm of the Radon-Nikodym cocycle denoted by ω : G × X → R, and given any ergodic flow R α (Y , η), we consider Orbit equivalence superrigidity for type III 0 actions 3 We prove in Proposition 3.4 that this action is ergodic and that its associated flow is given by the adjoint flow α, a new concept that we introduce in Definition 3.2. Since the adjoint operation is involutive, meaning that the adjoint of α is isomorphic to α, the skew product construction (1.1) provides a streamlined way of defining group actions with a prescribed associated flow.
The main result of this paper is the following OE-superrigidity theorem for actions of type III 0 . THEOREM A. Let G (X, μ) be a free, ergodic, non-singular action of type III 1 . Assume that G is finitely generated and has trivial center. Assume that the Maharam extension of G (X, μ) is simple and cocycle superrigid with countable targets. Denote by ω : G × X → R the logarithm of the Radon-Nikodym cocycle.
We provide a more precise version of Theorem A as Corollary 5.6 below. In this more precise version, the possible group actions 0 Z 0 with normal subgroup ¡ 0 that appear in the definition of OE superrigidity (v2) are explicitly described.
As we explain in Example 5.7 and Theorem 5.8, there are many concrete type III 1 actions G (X, μ) satisfying the assumptions of Theorem A. In particular, we obtain the following result.
Recall that given a commutative ring A and an integer n ≥ 2, the group E(n, A) is the subgroup of SL(n, A) generated by the elementary matrices having 1s on the diagonal and an element of A as an off-diagonal entry. For several rings, including Z[S −1 ] where S is a finite set of prime numbers and the ring of integers O K of an algebraic number field, we have that E(n, A) = SL(n, A) (see Example 6.8 for references and more examples). COROLLARY B. Let n ≥ 3 be an odd integer and let A ⊂ R be a subring containing an algebraic number that does not belong to Z. Assume that A is finitely generated as a ring. For every ergodic flow R α (Y , η), consider the action ( THEOREM C (1) Let n ≥ 3 be an integer, p a prime number and 0 < λ < 1. Consider the ring (2) Let G (X, μ) be any essentially free, ergodic, simple, type III 0 action. Then We prove Theorem C at the end of §4. For every free, ergodic, non-singular action G (X, μ) of a countable group G, can be entirely recovered from this group measure space construction L ∞ (X) G. This is a strictly stronger property than OE superrigidity, and both properties coincide if one can prove that M has a unique (group measure space) Cartan subalgebra; see, for example, [PV09]. When dealing with actions that are not measure-preserving, and especially with actions of type III 0 , the same nuances as with OE superrigidity appear and we get the natural definitions of W * -superrigidity (v1) and (v2).
For none of the concrete actions in Corollary B and Theorem C is it known whether the crossed product has a unique (group measure space) Cartan subalgebra, up to unitary conjugacy. Nevertheless, repeating the construction of [Vae13, Proposition D], we obtain ad hoc examples of group actions that are W * -superrigid (v2), of type III 0 , with a prescribed associated flow. We explain this in Remark 6.10.

Preliminaries
Recall that an action of a countable group G on a standard probability space (X, μ) is called non-singular if it preserves Borel sets of measure zero. We write (g · μ)(U ) = μ(g −1 · U ) and consider the Radon-Nikodym derivatives d(g · μ)/dμ, which are well defined almost everywhere. Given a non-singular action G (X, μ) of a countable group G on a standard probability space (X, μ), we consider the associated Maharam extension is the logarithm of the Radon-Nikodym 1-cocycle. We may and always will assume that ω is a strict 1-cocycle, meaning that the cocycle identity holds everywhere. We equip X × R with the G-invariant σ -finite measure dμ(x) × e −s ds. One considers the measure-scaling action Orbit equivalence superrigidity for type III 0 actions 5 π : X × R → Y . Since the actions of G and R on X × R commute, there is an essentially unique non-singular action R (Y , η) such that for all t ∈ R, we have that π(t · (x, s)) = t · π(x, s) for almost every (a.e.) (x, s) ∈ X × R. The action R (Y , η) is Krieger's associated flow of the action G (X, μ). By [Zim84, Proposition B.5], after discarding from (X, μ) a G-invariant Borel null set, we may assume that the factor map π : X × R → Y is strictly G-invariant and R-equivariant, that is, π(g · (x, s)) = π(x, s) and π(x, s + t) = t · π(x, s) for all g ∈ G, t ∈ R and (x, s) ∈ X × R. Writing ψ(x) = π(x, 0), we have found a Borel map ψ : X → Y satisfying for all x ∈ X, s ∈ R, g ∈ G.
Let G (X, μ) be an essentially free, ergodic, non-singular action of a countable group G on a non-atomic standard probability space (X, μ). Recall that the type of this action is defined as follows: if there exists a G-invariant probability measure ν ∼ μ, the action is of type II 1 ; if there exists a G-invariant infinite measure ν ∼ μ, the action is of type II ∞ ; in all other cases, the action is of type III. Also recall that G (X, μ) is of type II 1 or II ∞ if and only if the associated flow is isomorphic to the translation action R R. When the associated flow is not the translation action, there are three possibilities: if Y is reduced to one point, the action is said to be of type III 1 ; if the associated flow is isomorphic to the periodic flow R R/Z log λ with 0 < λ < 1, the action is said to be of type III λ ; finally, when the associated flow is properly ergodic, the action is said to be of type III 0 .
Two non-singular actions G (X, μ) and (Z, ζ ) are said to be conjugate if there exist an isomorphism of groups δ : G → and a non-singular isomorphism : (X, μ) → (Z, ζ ) such that (g · x) = δ(g) · (x) for all g ∈ G and a.e. x ∈ X. Two non-singular actions G (X, μ) and G (Z, ζ ) of the same group G are said to be isomorphic if there exists a non-singular isomorphism : (X, μ) → (Z, ζ ) such that (g · x) = g · (x) for all g ∈ G and a.e. x ∈ X. Two essentially free, ergodic, non-singular actions G (X, μ) and (Z, ζ ) are called stably orbit equivalent if there exist non-negligible Borel sets U ⊂ X, V ⊂ Z and a non-singular isomorphism : x ∈ U . The actions are called orbit equivalent if we may choose U = X and V = Z. When the actions are both of type II ∞ or type III, stable orbit equivalence is the same as orbit equivalence. Recall that the associated flow is invariant under stable orbit equivalence.
We say that a non-singular action G (X, μ) is induced if there exist a proper subgroup G 0 < G and a G 0 -invariant Borel set X 0 ⊂ X such that the sets (g · X 0 ) g∈G/G 0 are disjoint and μ(X \ G · X 0 ) = 0. We then say that G X is induced from G 0 X 0 . Given any non-singular action G 0 (X 0 , μ 0 ) and a larger countable group G containing G 0 , there is, up to isomorphism, a unique non-singular action G X that is induced from G 0 X 0 . Note that by construction, if G X is induced from G 0 X 0 , then G X and G 0 X 0 are stably orbit equivalent. For later reference, we record the following lemma.
it follows that X 0 is G 0 -invariant and that (g · X 0 ) g∈G/G 0 is a partition of X, up to measure zero. Since G X is not induced, it follows that G 0 = G and that X 0 = X, up to measure zero. This means that ϕ is essentially constant.
If G X is induced from G 0 X 0 , the map x → gG 0 for x ∈ g · X 0 is a G-equivariant map X → G/G 0 that is not essentially constant.
An essentially free, non-singular action (X, μ) is said to admit a fundamental domain if there exists a Borel set U ⊂ X such that all g · U , g ∈ , are disjoint and μ(X \ · U ) = 0. In that case, the quotient X/ is a well-defined standard measure space and identified with (U , μ).

Adjoint flows and type III 0 actions with prescribed associated flow
We say that a flow R (Z, ζ ) scales the σ -finite measure ζ if t · ζ = e t ζ for all t ∈ R.
Both the actions by (t, 0) and by (0, r) scale the measure ζ . By construction, is a non-singular ergodic action such that the actions of both R × {0} and {0} × R scale the measure ζ and such that R where ζ : R × Y → R is a 1-cocycle. Precomposing with the measure-preserving map Orbit equivalence superrigidity for type III 0 actions 7 and replacing ζ by a cohomologous 1-cocycle, we may assume that μ = μ. Expressing that the action of R × {0} scales the measure μ × γ gives us that ζ(t, y) = ω(t, y) + t. So we have found the required isomorphism.
Given the uniqueness of R 2 (Z, ζ ) in Proposition 3.1, we get the following well-defined notion of an adjoint flow and we automatically have that this adjoint is an involutive operation: the adjoint of α is isomorphic to α.
Definition 3.2. Given an ergodic flow R α (Y , η), the adjoint flow R α ( Y , η) is defined as the ergodic flow R Z/(R × {0}), where R 2 Z is the unique action given by Proposition 3.1.
Note that we can also define the adjoint flow α more concretely. Denoting by ω : R × Y → R the logarithm of the Radon-Nikodym cocycle of an ergodic flow R α Y , we consider the quotient (Y × R)/R, where R is acting by t · (y, s) = (t · y, t + ω(t, y) + s). On this quotient, we let R act by translation in the second variable. This is the adjoint flow α.
From this concrete description, it immediately follows that α ∼ = α whenever the flow α admits a finite or σ -finite equivalent R-invariant measure. In general, α need not be isomorphic to α, as the following example shows.
Example 3.3. Let R α (Y , η) be the ergodic flow given as the induction of an ergodic, type III 1 action Z α 0 (Y 0 , η 0 ). We prove that the adjoint flow α is not isomorphic to α.
Recall that the induced flow α is defined as follows. Consider the action R × Z R × Y 0 : (t, n) · (s, y) = (t − n + s, n · y). Then α is defined as the action of R on (R × Y 0 )/({0} × Z). To determine the adjoint flow α, denote by ω : Z × Y 0 → R the logarithm of the Radon-Nikodym cocycle for α 0 . Denote by λ the Lebesgue measure on R and define the measure γ such that (dγ /dλ)(t) = e −t . Then consider the action : (t, n, r) · (s, y, s ) = (t − n + s, n · y, t + ω(n, y) + r + s ).
The action of {0} × Z × {0} is measure-preserving. The actions of R × {(0, 0)} and {(0, 0)} × R are measure-scaling. By construction, α is given by We conclude that the adjoint flow α is given by By construction, the flow α comes with an R-equivariant map Y → R/Z. We prove that such an R-equivariant map does not exist for the adjoint flow α. Assuming the contrary, we find a map θ : R × Y 0 × R → R/Z that is invariant for the action of R × Z × {0} and that is equivariant for the action of {(0, 0)} × R. By the invariance under R × {(0, 0)} and the equivariance under {(0, 0)} × R, the map θ must be of the form θ(s, y, s ) = −s + s + ϕ(y) where ϕ : Y 0 → R/Z. The invariance under {0} × Z × {0} then says that ω(n, y) + ϕ(n · y) = ϕ(y) for all n ∈ Z and a.e. y ∈ Y 0 . This means that the map Y 0 × R → R/Z : (y, s) → s + ϕ(y) is invariant under the action Z Y 0 × R : n · (y, s) = (n · y, ω(n, y) + s). This action is ergodic because Z Y 0 is 8 S. Vaes and B. Verjans assumed to be of type III 1 . So the map (y, s) → s + ϕ(y) is essentially constant, which is absurd. So we have proven that α is not isomorphic to α.
PROPOSITION 3.4. Let G (X, μ) be a non-singular ergodic action of type III 1 , with logarithm of the Radon-Nikodym cocycle ω : G × X → R. Let R α (Y , η) be any ergodic flow. Then the action is ergodic and has associated flow α.
Proof. Let R 2 (Z, ζ ) be the unique action given by Proposition 3.1, associated with the ergodic flow R (Y , η). Consider the action The action of {e} × R scales the measure, while the action of G × {0} is measurepreserving. By definition, the action of G on It thus follows that the action in (3.2) is the Maharam extension of G X × Y together with its measure-scaling action of R. By the uniqueness of R 2 (Z, ζ ), we may as well identify where β : R × Y → R is the logarithm of the Radon-Nikodym cocycle for the adjoint flow R α Y . Then the action in (3.2) becomes · (x, s, y) = (g · x, ω(g, x) + β(r, y) + r + s, r · y).
Since G (X, μ) is ergodic and of type III 1 , the Maharam extension G X × R is ergodic. It follows that the G-invariant functions on X × R × Y are the functions that only depend on the Y -variable. Since R Y is ergodic, we conclude that the action in (3.2) is ergodic. We have proven that the action in (3.1) is ergodic and that its associated flow is identified with α : R Y .

Versions of OE superrigidity in the type III setting
Definition 4.1. We say that a free, ergodic, non-singular action G (X, μ) of a countable group G is simple if the action is not induced and if there are no non-trivial normal subgroups ¡ G for which (X, μ) admits a fundamental domain.
The motivation for this ad hoc notion of simplicity is the following. When G (X, μ) is induced from G 0 X 0 , we have a canonical stable orbit equivalence between G X and G 0 X 0 . When ¡ G is a normal subgroup such that (X, μ) admits a fundamental domain, we have a canonical stable orbit equivalence between G X and G/ X/ . So when G (X, μ) is not simple, there always is a certain absence of OE superrigidity and describing all stably orbit equivalent actions is necessarily cumbersome. For this reason, we mainly restrict ourselves to simple actions in this paper.
Recall that a non-singular action G (X, μ) of a countable group G on a standard probability space (X, μ) is called cocycle superrigid with countable target groups if every 1-cocycle : G × X → with values in a countable group is cohomologous to a group homomorphism δ : G → , viewed as a 1-cocycle that is independent of the X-variable.
Recall from the introduction the two versions (v1) and (v2)  Conversely, assume that : G × X → is a 1-cocycle with values in a countable group . Consider the free, non-singular, ergodic action By construction, this action is stably orbit equivalent to G X. Assume that this action is conjugate to an induction of G X. We have to prove that is cohomologous to a group homomorphism.
Take an injective group homomorphism δ : G → G × : δ(g) = (δ 1 (g), δ 2 (g)) and a measure space isomorphism : Z and is a conjugacy with respect to δ. Since the action of on X × admits a fundamental domain, the same is true for the action Ker δ 1 X. Since G X is simple, we find that δ 1 is faithful. Since Define the group homomorphism γ : We denote by θ its composition with ψ. By -equivariance, we get that θ(x, a) = θ 0 (x)a, where θ 0 : X → is a Borel map.

S. Vaes and B. Verjans
Expressing the G-equivariance gives us that so that is cohomologous to a group homomorphism.
As an essential ingredient to prove the first part of Theorem C, as well as to prove Corollary B, we need to establish cocycle superrigidity for linear and for affine actions on R n . We first recall the notion of essential cocycle superrigidity introduced in [DV21, Definition B]. We only formulate the version for connected Lie groups, which is the one that we need in this paper.
Lie group G, with universal cover π : G → G and = π −1 ( ), is said to be essentially cocycle superrigid with countable targets if for every 1-cocycle ω : × G → of the translation action G with values in a countable group , the lifted 1-cocycle ω : In [DV21, Propositions 4.1 and 4.2], it was proven that for every integer n ≥ 3, non-empty set of prime numbers S and real algebraic number field Q K ⊂ R with ring of integers O K , the dense subgroups SL(n, Z[S −1 ]) and SL(n, O K ) of SL(n, R) are essentially cocycle superrigid with countable targets. We prove the same result for a much larger family of dense subgroups of SL(n, R) and also for their corresponding subgroups of SL(n, R) R n .
For every integer n ≥ 2 and commutative ring A, we denote by SL(n, A) and GL(n, A) the groups of n × n matrices with entries in A and determinant 1, belonging to A * . Whenever 1 ≤ i, j ≤ n with i = j and a ∈ A, we denote by e ij (a) ∈ SL(n, A) the elementary matrix with 1s on the diagonal, a in position ij and 0s elsewhere. We denote by E(n, A) ⊂ SL(n, A) the subgroup generated by the elementary matrices. By Suslin's theorem (see [HOM89, Theorem 1.2.13]), for all n ≥ 3, E(n, A) is a normal subgroup of GL(n, A). For several rings A, it is known that E(n, A) = SL(n, A) (see Example 6.8 for references).
THEOREM 4.4. Let A ⊂ R be any countable subring containing an algebraic number that does not belong to Z. Let n ≥ 3 and let E(n, A) < < SL(n, A) be any intermediate subgroup.
(2) The dense subgroup A n < SL(n, R) R n is essentially cocycle superrigid with countable targets.
Before proving Theorem 4.4, we need two elementary lemmas, which are essentially contained in [Ioa14, Lemma 5.1].
LEMMA 4.5. Let G be a connected locally compact second countable group and let < G be a countable dense subgroup. Then the translation action G is ergodic and not induced.
Proof. Let 0 < be a subgroup and π : G → / 0 a Borel map such that π(gh) = gπ(h) for all g ∈ and a.e. h ∈ G. We have to prove that π is essentially constant. Since / 0 is countable, the set Since π(gh) = gπ(h) when g ∈ , the set U is essentially invariant under the diagonal translation action G × G. By continuity, U is essentially invariant under the diagonal translation action of G. We thus find a non-negligible Borel set V ⊂ G such that, up to measure zero, This means that π is essentially constant.
The following lemma is also essentially contained in [Ioa14, Lemma 5.1] and allows us to extend cocycle superrigidity from a subgroup 0 < to its normalizer N ( 0 ). Such a result goes back to [Pop05, Proposition 3.6].
LEMMA 4.6. Let (X, μ) be a free, ergodic, non-singular action and let : × X → be a 1-cocycle with values in a countable group . Let 0 < be a subgroup and let (X y , μ y ) y∈(Y ,η) be the ergodic decomposition of the action 0 (X, μ), with corresponding factor map π : factors through π for every g ∈ N ( 0 ).
Proof. Fix g ∈ N ( 0 ) and denote by α : 0 → 0 the automorphism α(h) = ghg −1 . Since for every h ∈ 0 , the function x → (h, x) factors through π , we find a measurable family (δ y ) y∈Y of group homomorphisms δ y : 0 → such that (h, x) = δ π(x) (h) for all h ∈ 0 and a.e. x ∈ X. Applying the 1-cocycle relation to gh = α(h)g gives us that is 0 -invariant. We can thus define the non-singular action N ( 0 ) (Y , η) such that π(g · x) = g · π(x) for all g ∈ N ( 0 ) and a.e. x ∈ X. We conclude that for a.e. y ∈ Y , we have (4.1) The action 0 (X y , μ y ) is ergodic and not induced. Also (4.1) is saying that the map X y → : is essentially constant on (X y , μ y ). Since this holds for a.e. y ∈ Y , we have proven that x → (g, x) factors through π .
Note that Lemma 4.5 also implies the following result, which we will use in combination with Lemma 4.6: if G is a locally compact second countable group and < G is a countable subgroup whose closure H = is connected, then the map π : G → H \G : π(g) = Hg realizes the ergodic decomposition of the translation action G and, for 12

S. Vaes and B. Verjans
every Hg ∈ H \G, the action of on π −1 (Hg) is isomorphic to the translation action H and thus not induced by Lemma 4.5.
Proof of Theorem 4.4. By our assumption, the ring A contains a rational number q ∈ Q \ Z or an irrational algebraic number α. In the first case, by taking a multiple of q, we find a prime p and a positive integer In the second case, we define the algebraic number field K = Q(α). Let d ≥ 2 be the degree of the minimal polynomial of α. The ring O K of integers of K is a finitely generated Z-module that is contained in K, which has {1, α, . . . , α d−1 } as a Q-vector space basis. We can thus take a positive integer We start by proving that E(n, A 0 ) < SL(n, R) is essentially cocycle superrigid with countable targets.
Using the real and complex embeddings of K, the group SL(n, O K ) is an irreducible lattice in a product of copies of SL(n, R) and SL(n, C); see, for example, [PR94, Theorems 5.7 and 7.12]. The groups SL(n, R) and SL(n, C) both have property (T); see, for example, [BHV08, Theorem 1.4.15]. Then the finite-index subgroup SL(n, A 0 ) of SL(n, O K ) is also such an irreducible lattice. In particular, SL(n, A 0 ) satisfies Margulis's normal subgroup theorem. Since E(n, A 0 ) is an infinite normal subgroup of SL(n, A 0 ), we conclude that E(n, A 0 ) < SL(n, A 0 ) has finite index. So E(n, A 0 ) is also an irreducible lattice in a product of copies of SL(n, R) and SL(n, C). By part 2 of [DV21, Theorem C] (and this is essentially [Ioa14, Theorem B]), we get that E(n, A 0 ) < SL(n, R) is essentially cocycle superrigid with countable targets.
We also need the following observation: given a countable dense subgroup of a connected Lie group G with universal cover π : G → G, the subgroup < G is essentially cocycle superrigid with countable targets if and only if the translation action of the dense subgroup = π −1 ( ) on G is cocycle superrigid with countable targets. One implication is obvious. So assume that < G is essentially cocycle superrigid with countable targets and that ω : × G → is a 1-cocycle with values in a countable group . Since the action of Ker π on G admits a fundamental domain, ω is cohomologous with a 1-cocycle ω 1 satisfying ω 1 (g, x) = e for all g ∈ Ker π and a.e. x ∈ G. This means that ω 1 (g, x) = ω 2 (π(g), π(x)) for all g ∈ and a.e. x ∈ G, where ω 2 : × G → is a 1-cocycle. By our assumption, ω 1 , and hence also ω, is cohomologous with a group homomorphism from to .
We then prove statement (1). Take E(n, A) < < SL(n, A). We write G = SL(n, R) and denote by π : G → G its universal cover. Since n ≥ 3, Ker π is a central subgroup of order 2 in G. Define = π −1 ( ). Let ω : × G → be a 1-cocycle, with lift ω : × G → . We have to prove that ω is cohomologous to a group homomorphism.
Since we have proven that E(n, A 0 ) < G is essentially cocycle superrigid, it follows from the observation above that ω is cohomologous with a 1-cocycle γ : × G → that has the property that x → γ (g, x) is essentially constant for every g ∈ π −1 (E(n, A 0 )).
Write [n] = {1, . . . , n}. For every k ∈ [n] and for every subring B ⊂ R, we define the following subgroups of E(n, B): and R k (R) respectively, and the latter are closed subgroups of G.
Since R is simply connected, there is for all i = j a unique continuous group homomorphism e ij : R → G such that π( e ij (t)) = e ij (t) for all t ∈ R. When i, j ∈ [n] \ {k} and s, t ∈ R, the image π([ e ik (t), e jk (s)]) of the commutator equals the identity element. Thus, [ e ik (t), e jk (s)] ∈ Ker π for all s, t ∈ R. By connectedness of R 2 , we find that e ik (t) commutes with e jk (s). There thus is a unique continuous group homomorphism For every k ∈ [n] and subring B ⊂ R, we denote by , it follows from Lemmas 4.5 and 4.6 that for every g ∈ C k (A), the map x → γ (g, x) factors through c k .
Take g ∈ π −1 (H k (A)C k (A)). Since g normalizes π −1 (C k (R)), the element g also normalizes its connected component of the identity C k (R). Since g normalizes π −1 (C k (A)) as well, it follows that g normalizes C k (A). Another application of Lemmas 4.5 and 4.6 then says that the map x → γ (g, x) factors through c k .
Fix i = j and fix g ∈ π −1 (e ij (A)). In the following paragraphs, we prove that x → γ (g, x) is essentially constant.
We have proven that the map x → γ (g, x) factors through c k for all k = i. This means that for all b = i, a = b and t ∈ R, we have γ (g, e ab (t)x) = γ (g, x) for a.e. x ∈ G. Since we can reason analogously using the subgroups R k , we also get for all a = j , b = a and t ∈ R that γ (g, e ab (t)x) = γ (g, x) for a.e. x ∈ G.
Define T = {h ∈ G | γ (g, hx) = γ (g, x) for a.e. x ∈ G }. Then T is a closed subgroup of G. Define T = π( T ). By the previous paragraph, e ab (R) ⊂ T when a = b, a = j or b = i. When a = j and b = i, we choose c ∈ [n] \ {i, j } and note that for all t ∈ R, e ac (t) e cb (1) e ac (−t) e cb (−1) = e ab (t), so that again e ab (R) ⊂ T . It follows that T = G. The closed subgroup T ⊂ G thus has index at most 2, so that T ⊂ G is open. Since G is connected, it follows that T = G. This means that x → γ (g, x) is essentially constant.
We have thus proven that x → γ (g, x) is essentially constant for every g ∈ π −1 (E(n, A)). Since π −1 (E(n, A)) is a normal subgroup of , a final application of Lemmas 4.5 and 4.6 implies that x → γ (g, x) is essentially constant for every g ∈ . This concludes the proof of statement (1).
To prove statement (2), we still write G = SL(n, R) with universal cover π : G → G. Note that G R n is the universal cover of G R n . Let ω : By the same argument as at the beginning of this proof, E(n, A 0 ) A n 0 is an irreducible lattice in a product of copies of SL(n, R) R n and SL(n, C) C n . By [BHV08, Corollary 1.4.16], these groups have property (T). By part 2 of [DV21, Theorem C], it follows that the dense subgroup E(n, A 0 ) A n 0 of SL(n, R) R n is essentially cocycle superrigid with countable targets.
By the observation at the beginning of the proof, we thus find that ω is cohomologous with a 1-cocycle γ such that x → γ (g, x) is essentially constant for every g ∈ π −1 (E(n, A 0 )) A n 0 . Denote by θ : G R n → G the natural quotient map. Since A n 0 is dense in R n and since A n commutes with A n 0 , it follows from Lemmas 4.5 and 4.6 that x → γ (a, x) factors through θ for all a ∈ A n . Since A n is a normal subgroup of A n , another application of Lemmas 4.5 and 4.6 implies that x → γ (g, x) factors through θ for all g ∈ A n . We thus find a Borel map γ 1 : ( A n ) × G → such that for all g ∈ A n and a.e. x ∈ G R n , we have γ (g, x) = γ 1 (g, θ(x)). From now on, we denote by g the elements of and we denote by a the elements of A n . The restriction of γ 1 to × G is a 1-cocycle for the translation action G. Above, we have proven that this action is cocycle superrigid with countable target groups. Choose a Borel map ϕ : G → and a group homomorphism δ : → such that γ 1 (g, y) = ϕ(gy) −1 δ(g) ϕ(y) for all g ∈ and a.e. y ∈ G. Replacing γ with the cohomologous 1-cocycle (g, x) → ϕ(θ(gx)) γ (g, x) ϕ(θ(x)) −1 , we may thus assume that γ 1 (g, y) = δ(g) for all g ∈ and a.e. y ∈ G. We denote δ y (a) = γ 1 (a, y) and note that δ y : A n → is a measurable family of group homomorphisms. To conclude the proof of the theorem, we have to show that for all a ∈ A n , the map y → δ y (a) is essentially constant.
When g ∈ , we have that π(g) ∈ SL(n, A n ) so that π(g)(a) ∈ A n . The group law in A n can then be expressed by g a = π(g)(a) g for all g ∈ , a ∈ A n . Applying the 1-cocycle relation for γ , we conclude that δ gy (π(g)a) = δ(g) δ y (a) δ(g) −1 for all g ∈ , a ∈ A n and a.e. y ∈ G. For every a ∈ A, denote e i (a) = ae i ∈ A n . It follows from (4.2) that for all g ∈ π −1 (L i (A)), we have δ gy (e i (a)) = δ(g) δ y (e i (a)) δ(g) −1 . By Lemma 4.5, for every fixed i ∈ [n] and a ∈ A, the map y → δ y (e i (a)) is invariant under left translation by the connected group π −1 (L i (R)), and thus of the form y → ζ(π(y) −1 (e i )) for some Borel map ζ : R n → . This means that we find measurable families (ρ i,z ) z∈R n of group homomorphisms ρ i,z : A → such that δ y (e i (a)) = ρ i,π(y) −1 (e i ) (a) for all i ∈ [n], a ∈ A and a.e. y ∈ G.
Since π(g) −1 (e i ) = e i and π(g) −1 (e j ) = e i + e j , we find that for all a ∈ A and a.e. y ∈ G. Since n ≥ 3, the map G → R n × R n : y → (π(y) −1 (e i ), π(y) −1 (e j )) is a non-singular factor map. We thus conclude that Denote by P the Polish group of Borel maps from R n to , where two such maps are identified if they are equal almost everywhere, where the topology is given by convergence in measure and where the group law is defined pointwise. For every v ∈ R n and F ∈ P, Fix a ∈ A and define F , G ∈ P by F (u) = ρ j ,u (a) and G(u) = ρ i,u (a) −1 . Then the map R n → P : v → F v G is continuous. By (4.3), this map takes values in the discrete subgroup < P of constant functions. Since R n is connected, it follows that v → F v is essentially constant. That means that we find group homomorphisms ρ j : A → such that ρ j ,u (a) = ρ j (a) for a.e. y ∈ R n . We conclude that for all j ∈ [n] and every a ∈ A, the map y → δ y (e j (a)) is essentially constant. So y → δ y (a) is also essentially constant for every a ∈ A n . This concludes the proof of the theorem.
The first part of Theorem C can now be immediately deduced from Theorem 4.4. We state and prove the following more general version.
COROLLARY 4.7. Let A ⊂ R be any countable subring containing an algebraic number that does not belong to Z. Let F ⊂ A * be a subgroup and n ≥ 3 an integer. Define < GL(n, A) as the group of matrices with det A ∈ F. Consider the action of G = A n on X = R n by (A, a) · x = A(a + x). The action G X is essentially free, ergodic, non-singular, simple and cocycle superrigid with countable targets. So the action is also OE superrigid (v1).
Denote by T the closure of {|a| | a ∈ F} in R * + . If T = R * + , the action is of type III 1 . If T = λ Z , the action is of type III λ . If T = {1}, the action is of type II ∞ .
Proof. Since A n is dense in R n , by Lemma 4.5, the action A n R n is ergodic and not induced. A fortiori, G R n is ergodic and not induced. Assume that ¡ G is a 16

S. Vaes and B. Verjans
normal subgroup whose action on R n admits a fundamental domain. Write 0 = ∩ A n . Let (A, a) ∈ . If A = 1, then for all b ∈ A n , Since 0 is globally invariant under SL(n, A), the closure of 0 in R n is a non-trivial closed subgroup of R n that is globally invariant under SL(n, R). So 0 is dense in R n . It follows that 0 R n is ergodic, contradicting the assumption that 0 R n admits a fundamental domain. So A = 1, and we have proven that ⊂ A n . If = {0}, we again find that is dense in R n . So is trivial, and we have proven that G R n is a simple action. Write G 0 = SL(n, A) A n . By Theorem 4.4, the dense subgroup G 0 of SL(n, R) R n is essentially cocycle superrigid with countable targets. By [DV21, Proposition 3.3], the action G 0 R n is cocycle superrigid with countable targets. As mentioned above, this action is ergodic and not induced. Since G 0 is a normal subgroup of G, it follows from Lemma 4.6 that G R n is also cocycle superrigid with countable targets. By Proposition 4.2, the action is also OE superrigid (v1).
The Maharam extension of G R n can be identified with the action G R n × R given by Since the translation action A n R n is ergodic, it follows that the G-invariant functions on R n × R are precisely the functions on R that are invariant under translation by all log | det A|, A ∈ GL(n, A), det A ∈ F. So these are the functions on R that are invariant under translation by {log |a| | a ∈ F}, so that the type of G R n is as described in the corollary.
PROPOSITION 4.8. Let G (X, μ) be a free, ergodic, non-singular action of type III 0 . If G X is not induced, then G X is not cocycle superrigid with countable targets. If G X is simple, then G (X, μ) is not OE superrigid (v1).
Proof. Let G (X, μ) be a free, ergodic, non-singular action of type III 0 . Assume that every 1-cocycle : G × X → Z is cohomologous to a group homomorphism. We prove that G X must be an induced action. By Proposition 4.2, this suffices to prove the proposition.
Note that π 1 is not essentially constant, since otherwise π(x) takes values in a countable set for a.e. x ∈ X, contradicting π * (μ) ∼ η. We can then choose k ∈ N large enough such that, denoting by ψ : Z 2 → Z/2 k Z the canonical quotient map, θ = ψ • π is not essentially constant. Since θ(g · x) = δ(g) + θ(x) for all g ∈ G and a.e. x ∈ X, it follows that the action G (X, μ) is induced in a non-trivial way.
Proof of Theorem C. This now follows immediately from Corollary 4.7 and Proposition 4.8.

Cocycle and OE superrigidity for actions of type III 0
Let G (X, μ) be a non-singular action, with logarithm of the Radon-Nikodym 1-cocycle ω : G × X → R. Consider the Maharam extension G X × R, together with the commuting measure-scaling action R X × R (see (2.1) and (2.2)). Denote by R (Y , η) the associated flow. As explained at the start of §2, we have a strictly G-invariant and R-equivariant Borel map π : X × R → Y and we define ψ : X → Y by (2.3).
Given a subgroup 0 < , we denote by C ( 0 ) = {g ∈ | gh = hg for all h ∈ 0 } the centralizer of 0 inside . THEOREM 5.1. Let G (X, μ) be a non-singular action of a countable group G on a standard probability space (X, μ). Let ω : G × X → R be the logarithm of the Radon-Nikodym cocycle and let G ( X, μ) be the Maharam extension, with ergodic decomposition ( X y , μ y ) y∈Y , associated flow R (Y , η) and Borel map ψ : X → Y as in (2.3).
Let be a countable group. Assume that G is finitely generated and that for η-a.e. y ∈ Y , the ergodic action G ( X y , μ y ) is not induced and cocycle superrigid with target . Then for any 1-cocycle : G × X → , there exist a group homomorphism δ : G → and a strict 1-cocycle γ : R × Y → C (δ(G)) such that is cohomologous with the 1-cocycle Proof. Define the 1-cocycle : (G × R) × X → : ((g, t), (x, s)) = (g, x). First restrict to a 1-cocycle for the action G X with ergodic decomposition given by π : X → Y . As explained in detail in [FMW04], we may consider as a measurable family ( y ) y∈Y of 1-cocycles for the measurable family of actions G ( X y , μ y ). By assumption, η-a.e. y is cohomologous to a group homomorphism δ y : G → .
(5.2) So γ 0 : R × Y → is a 1-cocycle. Then (5.1) is saying that for all t ∈ R, g ∈ G, we have Since G is finitely generated, the set of group homomorphisms from G to is countable. We thus find a group homomorphism δ : G → such that δ y = δ for all y in a non-negligible Borel subset U ⊂ Y . Combining (5.3) with the ergodicity of R (Y , η), it follows that δ y is conjugate to δ for η-a.e. y ∈ Y . We then find a Borel map ρ : Y → so that δ y (g) = ρ(y) −1 δ(g) ρ(y) for η-a.e. y ∈ Y and all g ∈ G. Replacing ϕ(x, s) by ρ(π(x, s))ϕ(x, s), we may assume that δ y = δ for η-a.e. y ∈ Y .
as in (1.1). If G is finitely generated and if the Maharam extension of G (X, μ) is not induced and cocycle superrigid with countable targets, then every 1-cocycle : G × X × Y → for the action (5.4) with values in a countable group is cohomologous with a 1-cocycle of the form where δ : G → is a group homomorphism and γ : R × Y → C (δ(G)) is a 1-cocycle.
Proof. As in the proof of Proposition 3.4, we take the unique action R 2 (Z, ζ ) given by Proposition 3.1, associated with the ergodic flow R (Y , η). Identify Y = Z/({0} × R) and denote by π 1 : Z → Y the corresponding factor map. Also write Y = Z/(R × {0}) and denote by π 2 : Z → Y the corresponding factor map. The Maharam extension of G X × Y together with its measure-scaling action of R is then given by , (ω(g, x), t) · z).
Remark 5.4. When G (X, μ) is a free, ergodic, non-singular action of type III, with Maharam extension G ( X, μ) whose ergodic decomposition is denoted by ( X y , μ y ) y∈Y , it follows from [Tak03, Theorem XII.1.1] that for a.e. y ∈ Y , the action G ( X y , μ y ) is of type II ∞ .
To give examples where Theorem 5.1 applies, we thus need cocycle superrigidity for concrete actions of type II ∞ . For the specific actions appearing in Theorem 5.3, by construction, the actions G ( X y , μ y ) are almost everywhere the same.
Both Theorems 5.1 and 5.3 immediately lead to OE-superrigidity results. We start with the following result.
COROLLARY 5.5. Let G be a finitely generated group with trivial center and let G (X, μ) be an essentially free, non-singular, ergodic action. Let ω : G × X → R be the logarithm of the Radon-Nikodym cocycle and let G ( X, μ) be the Maharam extension, with associated flow R (Y , η) and Borel map ψ : X → Y as in (2.3). Assume that for η-a.e. y ∈ Y , the action G ( X y , μ y ) is simple and cocycle superrigid with countable target groups.
Then G (X, μ) satisfies the OE-superrigidity property (v2) defined in the introduction.
More precisely, any free non-singular ergodic action that is stably orbit equivalent to G (X, μ) is conjugate to an induction of an action of the form (g, x), ψ(x))ba −1 ), where is a countable group and γ : R × Y → is a strict 1-cocycle.
Proof. Let (Z, ζ ) be a free, ergodic, non-singular action and let : U ⊂ X → Z be a stable orbit equivalence between G X and Z. By ergodicity of G (X, μ), we can choose a Borel map θ : X → G such that θ(x) = e for all x ∈ U and θ(x) · x ∈ U for a.e. x ∈ X. Define 0 : X → Z : 0 (x) = (θ (x) · x). We then define the Zimmer 1-cocycle : G × X → such that 0 (g · x) = (g, x) · 0 (x) for all g ∈ G and a.e.
To translate the cocycle superrigidity Theorem 5.1 to an OE-superrigidity theorem, we use the connection with measure equivalence as developed in [Fur98,§3] (see also [DV21, Lemma 2.2] for a result that exactly suits our purposes). Define the action which commutes with the right translation action by in the second variable. By the results cited above, the action G X × admits a fundamental domain and there is a natural isomorphism of -actions α : G\(X × ) → Z with the property that (x) ∈ · α(x, e) for a.e. x ∈ U . By Theorem 5.1, we find a group homomorphism δ : G → and a 1-cocycle γ : R × Y → C (δ(G)) such that is cohomologous with the 1-cocycle Let ϕ : X → be a Borel map such that 1 (g, implements an isomorphism between the action G X × in (5.6) and the action Moreover, the action and the isomorphism commute with the -action. We thus still find an isomorphism of -actions α 1 : G\(X × ) → Z with the property that (x) ∈ · α 1 (x, e) for a.e. x ∈ U . The 1-cocycle 1 : G × X : 1 (g, (x, s)) = 1 (g, x) for the Maharam extension G X is, by construction, cohomologous with the 1-cocycle (g, (x, s)) → δ(g). Since the action G X × admits a fundamental domain, a fortiori, the same holds for the action G X × : g · (x, s, b) = (g · (x, s), δ(g)b), and thus for the action Ker δ X. Since we assumed that a.e. action G X y is simple, the normal subgroup Ker δ must be trivial. So δ : G → is faithful. Define = C (δ(G)). Since G has trivial center, δ(G) ∩ = {e}. We have thus found a subgroup δ(G) × < . Since 1 takes values in δ(G) × , the action Under the natural identification G\(X × δ(G) × ) = X × and the isomorphism δ × id : G × → δ(G) × , this last action is precisely the action given by (5.5).
In exactly the same way as Theorem 5.3 is deduced from Theorem 5.1, we can deduce the following result from Corollary 5.5. We thus omit the proof. Note that Theorem A is contained in the following corollary.
COROLLARY 5.6. Let G (X, μ) be a free, ergodic, non-singular action of type III 1 . Assume that G is finitely generated and has trivial center. Assume that the Maharam as in (1.1). Then G (X × Y , μ × η) satisfies the OE-superrigidity property (v2) defined in the introduction.
More precisely, any free non-singular ergodic action that is stably orbit equivalent to the action (5.8) is conjugate to an induction of an action of the form γ (ω(g, x), y)ba −1 ), where is a countable group and γ : R × Y → is a 1-cocycle.
There are several concrete group actions that satisfy the assumptions of Corollary 5.6. We start with the following example of [PV08].
Example 5.7. Whenever n ≥ 5 is an odd integer and G < SL(n, R) is a lattice, the action G R n /R * + satisfies the hypotheses of Corollary 5.6. Indeed, by [PV08, Theorem 1.3], the Maharam extension G R n is cocycle superrigid. Moreover, these groups G have property (T), so that they are finitely generated. Since n is odd, G has trivial center. By [PV08, Lemmas 5.6 and 6.1], the action G R n is doubly ergodic and not induced. By Margulis's normal subgroup theorem, a normal subgroup of G is either trivial (because n is odd) or of finite index, and thus acting ergodically on R n . So the action G R n is simple.
In the following theorem, we prove that all the assumptions of Corollary 5.6 are satisfied for the action G R n /R * + when G ranges over a broad family of dense subgroups SL(n, R) and n ≥ 3 is an odd integer. This then leads to the proof of point (1) in Corollary B (see Corollary 6.7 below).
THEOREM 5.8. Let A ⊂ R be any countable subring containing an algebraic number that does not belong to Z. Let n ≥ 3 be an integer and E(n, A) < G < SL(n, A). Then the linear action G R n is essentially free, ergodic, non-singular, not induced and cocycle superrigid with countable targets. If n is odd, then G has trivial center and the action is simple. If A is finitely generated as a ring, then E(n, A) is a finitely generated group.
Proof. Since G < SL(n, R) is dense, it follows from Lemma 4.5 that the action G R n is ergodic and not induced. By density of G < SL(n, R), the center of G belongs to the center of SL(n, R), which is trivial if n is odd. If n is odd and ¡ G is a normal subgroup whose action on R n admits a fundamental domain, then the closure of in SL(n, R) is, by density of G, a normal subgroup of SL(n, R). Since n is odd, it follows that either = {1} or = SL(n, R). In the second case, is a dense subgroup of SL(n, R), so that R n is ergodic. It thus follows that = {1}.
By Theorem 4.4, G < SL(n, R) is essentially cocycle superrigid with countable targets. By [DV21, Proposition 3.3], the action G R n is cocycle superrigid with countable targets.
By [HOM89,Proposition 4.3.11], the group E(n, A) is finitely generated when n ≥ 3 and A is finitely generated as a ring.

Conjugacy and classification results
In Corollary 5.5, we proved that free, non-singular, ergodic actions G (X, μ) with a sufficiently rigid Maharam extension G ( X, μ) satisfy the OE-superrigidity property (v2) and we described all stably orbit equivalent actions. We now prove the following complete classification up to conjugacy of this class of stably orbit equivalent actions.
For the formulation of the following proposition, note that every conjugacy of actions (and actually every stable orbit equivalence) gives rise to a canonical associated isomorphism between the associated flows.
PROPOSITION 6.1. Let G (X, μ) be an essentially free, non-singular, ergodic action. Make the same assumptions as in Corollary 5.5. Whenever γ : R × Y → is a strict 1-cocycle with values in a countable group and whenever σ : G × → is a faithful group homomorphism to a countable group , we denote by β(γ , σ ) the -action defined as the induction of the action along the embedding σ : G × → .
(1) An essentially free, non-singular, ergodic action is stably orbit equivalent to G (X, μ) if and only if it is conjugate to β(γ , σ ) for some γ , σ as above.
Point (1) of Proposition 6.1 is just a repetition of Corollary 5.5. We deduce point (2) of Proposition 6.1 from the following two results. First, in Proposition 6.2, we describe when and how an action of the form β(γ ) in (6.1) is induced. Second, in Proposition 6.3, we prove when two actions of the form β(γ ) are conjugate.
For our main family of group actions G X × Y defined in (1.1), it then remains to analyze when two such actions are conjugate. Under the appropriate assumption, we prove in Proposition 6.4 that this happens if and only if the G-actions G X are conjugate and the flows R Y are isomorphic. In particular, we find the outer automorphism groups of these type III 0 orbit equivalence relations.
Before proving Proposition 6.1, we clarify the following subtle point. When γ and γ are strict 1-cocycles that are cohomologous, expressed by the almost everywhere S. Vaes and B. Verjans equality γ (t, y) = ϕ(t · y) γ (t, y) ϕ(y) −1 , there is a natural isomorphism γ ,γ between the actions β(γ ) and β(γ ). This follows immediately by observing that the Maharam extension of β(γ ), together with its commuting R-action, is isomorphic to Moreover, the map Then the map (x, s, b) → (x, s, ϕ(π(x, s))b) is a well-defined isomorphism between the Maharam extensions. Taking the quotient by the action of R, we find γ ,γ . PROPOSITION 6.2. Let G (X, μ) be an essentially free, non-singular, ergodic action with Maharam extension G ( X, μ) and associated flow R (Y , η). Assume that for η-a.e. y ∈ Y , the action G ( X y , μ y ) is not induced. Let γ : R × Y → be a strict 1-cocycle with values in a countable group .
Proof. If γ takes values in 0 , we have by construction that β(γ ) is induced from G × 0 acting on X × 0 . So we only prove the converse and assume that β(γ ) is induced from Z. Write I = (G × )/ . Consider the Maharam extension of β(γ ) given by (6.2). Since β(γ ) is induced from Z, we find a Borel map θ : X × R × → I that is R-invariant and (G × )-equivariant.
Since for a.e. y ∈ Y , we have that G X y is not induced, the G-equivariance of θ implies that θ(x, s, b) = θ 1 (π(x, s), b), where θ 1 : Y × → I . In particular, θ is a G-invariant map. It follows that G × {e} ⊂ , so that = G × 0 for some subgroup 0 < . From now on, we identify I = / 0 , on which G acts trivially. The -equivariance of θ implies that θ 1 is also -equivariant and thus of the form θ 1 (y, b) = b −1 θ 2 (y) 0 for some Borel map θ 2 : Y → . Expressing the R-invariance of θ and thus the invariance of θ 1 under the action t · (y, s) = (t · y, γ (t, y)s), we find that This means precisely that the cohomologous 1-cocycle γ 0 (t, y) = θ 2 (t · y) −1 γ (t, y)θ 2 (y) takes values in 0 . Using the notation introduced before the proposition, this also means that γ 0 ,γ (Z) = X × 0 . PROPOSITION 6.3. For i ∈ {1, 2}, let G i (X i , μ i ) be essentially free, non-singular, ergodic actions with Maharam extension G i ( X i , μ i ) and associated flow R (Y i , η i ). Assume that for η i -a.e. y ∈ Y i , the action G i ( X i,y , μ i,y ) is simple. Assume that the groups G i have trivial center. Let γ i : R × Y i → i be strict 1-cocycles.
We finally turn to our main family of group actions, given by (1.1). For i ∈ {1, 2}, let G i (X i , μ i ) be non-singular actions and denote by ω i : G i × X i → R the logarithm of the Radon-Nikodym cocycle. Let R (Y i , η i ) be non-singular flows. Consider the actions If 1 : X 1 → X 2 is a conjugacy between the actions G 1 X 1 and G 2 X 2 and if 2 : Y 1 → Y 2 is an isomorphism of the flows, writing ρ(x) = log(d(( 1 ) * μ 1 )/dμ 2 ), the map is a conjugacy between σ 1 and σ 2 . We now prove in Proposition 6.4 that under the appropriate assumption, the converse also holds. In Lemma 6.6, we explain that this assumption indeed holds for actions of the form G R n /R * + when G < SL(n, R) is a countable dense subgroup. This then leads to a proof of Corollary B. PROPOSITION 6.4. For i ∈ {1, 2}, let G i (X i , μ i ) be essentially free, ergodic, non-singular actions of type III 1 . Assume that the Maharam extensions G i ( X i , μ i ) admit a unique measure-scaling action commuting with the G i -action. Let R (Y i , η i ) be ergodic flows.
Every conjugacy between the actions σ i in (6.4) is of the form (6.5) for a conjugacy 1 between G 1 X 1 and G 2 X 2 and an isomorphism of flows 2 : Y 1 → Y 2 .
Proof. Denote by R 2 (Z i , ζ i ) the unique actions associated with the flows R (Y i , η i ) given by Proposition 3.1, together with the factor maps π i : Z i → Y i satisfying π i ((t, r) · z) = t · π i (z). We realize the Maharam extension of σ i together with its measure-scaling action as , (ω(g, x), r) · z). (6.6) We have the canonical factor map i : Assume that δ : G 1 → G 2 is a group isomorphism and : X 1 × Y 1 → X 2 × Y 2 is a δ-conjugacy between σ 1 and σ 2 . Denote by : X 1 × Z 1 → X 2 × Z 2 the canonical measure-preserving lift, which is a (δ × id)-conjugacy for the actions σ i in (6.6) and which satisfies 2 • = • 1 .
Define the action γ : . Note that γ commutes with σ i . We claim that is automatically γ -equivariant.
To prove this claim, we temporarily identify Z i = R × Y i with the action R 2 Z i given by (t, r) · (s, y) = (t + r + β(r, y) + s, r · y). Under this identification, : Since G i X i is of type III 1 , the Maharam extension G i X i × R is ergodic. Therefore, must be of the form (x, s, y) = (θ y (x, s), ( y)), where : Y 1 → Y 2 is a non-singular isomorphism and, for a.e. y ∈ Y 1 , the map θ y : Since is measure-preserving, a.e. θ y is measure-scaling. We assumed that these Maharam extensions admit a unique commuting measure-scaling action. It follows that θ y is equivariant with respect to translation in the second variable. This means that is equivariant with respect to translation in the second variable. Thus, the claim is proven.
We thus consider the actions G i × R 2 X i × Z i : (g, t, r) · (x, z) = (g · x, (ω(g, x) + t, r) · z) and we have proven that is a (δ × id)-conjugacy between these actions. Since the action of R 2 on Z i is ergodic, this forces to be of the form (x, z) = ( 1 (x), x (z)), where 1 : X 1 → X 2 is a δ-conjugacy and, for a.e. x ∈ X 1 , x : Z 1 → Z 2 is an isomorphism between the actions R 2 Z i . Define ρ(x) = log(d(( 1 ) * μ 1 )/dμ 2 ) and denote x (z) = (ρ( 1 (x)), 0) · x (z). We still have that x is an isomorphism between the actions R 2 Z i . With respect to the measures μ i × ζ i on X i × Z i , the isomorphism is measure-preserving. It then follows that a.e. x is measure-preserving with respect to the measures ζ i on Z i .
Recall that a unitary representation π : G → U (H ) of a locally compact group G is said to be a C 0 -representation if for every ε > 0 and ξ , η ∈ H , there exists a compact subset K ⊂ G such that | π(g)ξ , η | < ε for all g ∈ G \ K.
LEMMA 6.5. Let n ≥ 1 be an integer. Write G = GL(n, R) R n . Then the unitary representation π : G → U (L 2 (R n )) : (π(A, a) In particular, the action G R n : (A, a) · x = A(a + x) induces a homeomorphism of G onto a closed subgroup of the Polish group of non-singular automorphisms of R n with the Lebesgue measure.
Proof. Assume the contrary. We then find ξ , η ∈ L 2 (R n ), ε > 0 and a sequence (A k , a k ) ∈ G that tends to infinity in G such that | π(A k , a k )ξ , η | > ε for all k.
We view GL(n, R) as a subgroup of G and we denote by D n < GL(n, R) the subgroup of diagonal matrices with positive real numbers on the diagonal. Since GL(n, R) = O(n, R)D n O(n, R), we can write (A k , a k ) = g k (d k , b k )h k with g k , h k ∈ O(n, R), d k ∈ D n and b k ∈ R n . After passing to a subsequence, we may assume that g k and h k converge to g and h, respectively. Replacing ξ by π(h)ξ and replacing η by π(g) * η, we may then further assume that | π(d k , b k )ξ , η | > ε for all k.
To reach a contradiction, it thus suffices to prove that the representation θ : R * which is compact.
LEMMA 6.6. For i = 1, 2 and integers n i ≥ 2, let G i < SL(n i , R) be dense subgroups and consider the actions G i R n i . If : R n 1 → R n 2 is a δ-conjugacy between these actions, we have n 1 = n 2 and there is a unique A ∈ GL(n 1 , R) such that (x) = A(x) for a.e. x ∈ R n 1 and δ(g) = AgA −1 for all g ∈ G 1 .
In particular, if n ≥ 2 and G < SL(n, R) is a dense subgroup, then the action R R n : t · x = e −t/n x is the unique measure-scaling action that commutes with G R n .
Proof. Denote by Aut(R n i ) the Polish group of non-singular automorphisms of R n i . By Lemma 6.5, we may view SL(n i , R) as a closed subgroup of Aut(R n i ). By our assumption, G 1 −1 = G 2 . Taking the closure in Aut(R n i ), we find that SL(n 1 , R) −1 = SL(n 2 , R). This means in particular that δ extends to a group isomorphism and homeomorphism δ : SL(n 1 , R) → SL(n 2 , R). It follows that n 1 = n 2 and we write n = n 1 = n 2 . Write G = SL(n, R) and denote by H < G the closed subgroup fixing the first basis vector e 1 . We view as a δ-conjugacy for the transitive action G G/H. This means that δ(H) = g 0 Hg −1 0 for some g 0 ∈ G and (gH) = δ(g)g 0 H for a.e. g ∈ G. Since δ is an automorphism of SL(n, R), we find B ∈ GL(n, R) such that either δ(g) = BgB −1 for all g ∈ G, or δ(g) = B(g −1 ) T B −1 for all g ∈ G. In the second case, δ(H) and H are not conjugate. So we are in the first case. We get that g 0 = BC where C ∈ GL(n, R) normalizes H. That means that Ce 1 = ae 1 for some a ∈ R * . Translating the formula (gH) = δ(g)g 0 H to R n , we have proven that (x) = aB(x) for a.e. x ∈ R n . Writing A = aB, the first part of the lemma is proven.
In particular, if G < SL(n, R) is a dense subgroup, then the only non-singular automorphisms of R n that commute with G R n are given by x → ax for some a ∈ R * . This transformation scales the measure by |a| n . The only measure-scaling action R R n commuting with G R n is thus given by t · x = e −t/n x.
We have now gathered enough material to prove Corollary B. We also add the computation of the outer automorphism group of these type III 0 orbit equivalence relations. For every ergodic flow R α (Y , η), we denote by Aut R (α) the Polish group of all non-singular automorphisms of (Y , η) that commute with the flow α. Note that R ⊂ Aut R (α) by definition. ( 6 . 1 0 ) Proof. By Theorem 5.8, under the hypotheses of Corollary B, the actions E(n, A) R n /R * + satisfy all the assumptions of Corollary 5.6. So by Corollary 5.6, the actions β(n, A, α) are essentially free, ergodic, simple and OE superrigid (v2), with associated flow α.
In particular, if β(n, A, α) and β(n , A , α ) are stably orbit equivalent, the actions must be conjugate. By Proposition 6.4 and Lemma 6.6, the flows α and α are isomorphic and the actions E(n, A) R n /R * + and E(n , A ) R n /R * + are conjugate. Then their Maharam extensions E(n, A) R n and E(n , A ) R n are also conjugate. By Lemma 6.6, n = n and there is an A ∈ GL(n, R) such that AE(n, A)A −1 = E(n, A ).
For every subring A 1 ⊂ R, we denote by M(n, A 1 ) the ring of n × n matrices with entries in A 1 . We claim that the subring N(n, A 1 ) ⊂ M(n, A 1 ) generated by E(n, A 1 ) is equal to M(n, A 1 ). For all i, j ∈ {1, . . . , n} and a ∈ A 1 , we denote by E ij (a) the matrix that has the entry a in position ij and 0s elsewhere. When i = j and a ∈ A 1 , we have that 1 + E ij (a) ∈ E(n, A 1 ) and 1 ∈ E(n, A 1 ). Thus, E ij (a) ∈ N(n, A 1 ). Since SL(n, Z) = E(n, Z) ⊂ E(n, A 1 ), given i = j , the matrix σ ij with entry 1 in position ij , entry −1 in position ji, and 0s elsewhere belongs to E(n, A 1 ). Thus, also E ii (a) = σ ij E ji (a) ∈ N(n, A 1 ). This proves the claim.
Since AE(n, A)A −1 = E(n, A ), the claim above implies that AM(n, A)A −1 = M(n, A ). So for every i ∈ {1, . . . , n} and a ∈ A, we have that AE ii (a)A −1 ∈ M(n, A ). Taking the jj -entry, it follows that A ji (A −1 ) ij a ∈ A . Summing over i, it follows that a ∈ A . So A ⊂ A . By symmetry, also the converse inclusion holds, so that A = A .
The arguments above apply in particular to the self orbit equivalences of β(n, A, α) and give us that Out(R(n, A, α)) ∼ = N GL(n,R) (E(n, A)) R * + E(n, A) × Aut R (α).
When A = Z[S −1 ], every fractional ideal of A is principal, so that N GL(n,K) (E(n, A)) = K * GL(n, A). By [HOM89, Theorem 4.3.9], we also have that E(n, A) = SL(n, A). So the natural map GL(n, A) → N GL(n,K) (E(n, A)) K * E(n, A) is surjective and has kernel {A ∈ GL(n, A) | det A ∈ (A * ) n }. The group of units of A = Z[S −1 ] is the free abelian group generated by p ∈ S, so that combining (6.11) en (6.12), we have proven (6.9).
When K is an algebraic number field with [K : Q] ≥ 2 and A = O K , again by [HOM89, Theorem 4.3.9], we have that E(n, A) = SL(n, A). Denote by I (K) the group of fractional ideals in K. As mentioned in [OM65, Example 6.5], the following two statements hold. If A ∈ GL(n, K), a ∈ I (K) and A(A n ) = aA n , we have a n = det(A) A. Conversely, if a ∈ I (K), α ∈ K * and a n = α A, there exists an A ∈ GL(n, K) such that A(A n ) = aA n and det A = α. For completeness, we provide a more detailed argument. The first implication is contained in [OM73,81:7]. For the second implication, take a ∈ I (K) and α ∈ K * with a n = α A. Denote by e 1 , . . . , e n the standard basis of K n . By [OM73,81:5], we find b ∈ I (K) and B ∈ GL(n, K) such that aA n = B(be 1 + Ae 2 + · · · + Ae n ). By our assumption and [OM73, 81:7], we have α A = a n = det(B) b. Denote by D ∈ GL(n, K) the diagonal matrix with D 11 = α det(B) −1 and D ii = 1 for i = 1. Since b = D 11 A, writing A = BD, we conclude that A(A n ) = aA n . By construction, det(A) = α.
Define the subgroup X n (K) of the abelian group I (K) × K * by X n (K) = {(a, α) ∈ I (K) × K * | a n = αA}. Define the subgroup Y n (K) ⊂ X n (K) by Y n (K) = {(βA, β n ) | β ∈ K * }. Define V n (K) = X n (K)/Y n (K). To conclude the proof of the corollary, we prove the following two statements: The projection on the first coordinate gives a surjective group homomorphism V n (K) → Cl n (K) with kernel A * /(A * ) n . We prove that this homomorphism is split. Since Cl n (K) is a finite abelian group in which the order of every element divides n, it suffices to prove that every element of order k | n in Cl n (K) can be lifted to an element of order k in V n (K). When a ∈ I (K) and a k = βA with β ∈ K * , write n = km and note that (a, β m ) defines such a lift of order k. So V n (K) ∼ = A * /(A * ) n × Cl n (K). For every (a, α) ∈ X n (K), by the discussion above, we can choose A ∈ GL(n, K) such that A(A n ) = aA n and det A = α. This matrix A is uniquely determined up to right multiplication by a matrix in SL(n, A). This realizes a surjective group homomorphism X n (K) → N GL(n,K) (SL(n, A)) K * SL(n, A) and the kernel of this homomorphism is by construction equal to Y n (K). This concludes the proof of the corollary.
Example 6.8. Let A = Z[S −1 ] where S is a finite non-empty set of prime numbers, or let A = O K be the ring of integers of an algebraic number field K ⊂ R with [K : Q] ≥ 2. As mentioned above, we have E(n, A) = SL(n, A) for all n ≥ 3, and a proof can, for instance, be found in [HOM89, Theorem 4.3.9]. By [Sus77, Corollaries 6.6 and 7.10], for the same rings A as in the previous paragraph and for all integers 0 ≤ k ≤ s, we also have that E(n, A 1 ) = SL(n, A 1 ) when A 1 is the ring of (Laurent) polynomials over A, defined by A 1 = A[X 1 , . . . , X k , X ±1 k+1 , . . . , X ±1 s ], finitely generated and has trivial center, by Corollary 5.6, it suffices to prove that the action G R 5 × [0, 1] G : g · (x, z) = (π(g)x, g · z) is simple and cocycle superrigid with countable targets. Simplicity is easy to check and cocycle superrigidity was proven in [Vae13,Proposition D].