Proof. If $G \curvearrowright (X,\mu )$ is not induced, $G \curvearrowright I$ and $\varphi : X \to I$ is G-equivariant, we can take $i_0 \in I$ such that $X_0 = \{x \in X \mid \varphi (x) = i_0\}$ is non-negligible. Defining $G_0 = \{g \in G \mid g \cdot i_0 = i_0\}$ , it follows that $X_0$ is $G_0$ -invariant and that $(g \cdot X_0)_{g \in G/G_0}$ is a partition of X, up to measure zero. Since $G \curvearrowright X$ is not induced, it follows that $G_0 = G$ and that $X_0 = X$ , up to measure zero. This means that $\varphi $ is essentially constant.

If $G \curvearrowright X$ is induced from $G_0 \curvearrowright X_0$ , the map $x \mapsto gG_0$ for $x \in g \cdot X_0$ is a G-equivariant map $X \to G/G_0$ that is not essentially constant.

Proof. Denote by $\omega : \mathbb {R} \times Y \to \mathbb {R}$ the logarithm of the Radon–Nikodym cocycle. Define the measure $\gamma $ on $\mathbb {R}$ by $d\gamma (t) = e^{-t} \, dt$ . Define $(Z,\zeta ) = (Y \times \mathbb {R},\eta \times \gamma )$ and define the action

$$ \begin{align*}\mathbb{R}^2 \curvearrowright (Z,\zeta) : (t,r) \cdot (y,s) = (t \cdot y, \omega(t,y) + t + r + s).\end{align*} $$

Both the actions by $(t,0)$ and by $(0,r)$ scale the measure $\zeta $ . By construction, $Z/(\{0\} \times \mathbb {R}) = Y$ .

Now assume that $\mathbb {R}^2 \curvearrowright (Z',\zeta ')$ is a non-singular ergodic action such that the actions of both $\mathbb {R} \times \{0\}$ and $\{0\} \times \mathbb {R}$ scale the measure $\zeta '$ and such that $\mathbb {R} \curvearrowright Z'/(\{0\} \times \mathbb {R})$ is isomorphic to $\mathbb {R} \curvearrowright Y$ . We prove that $\mathbb {R}^2 \curvearrowright (Z',\zeta ')$ is isomorphic to $\mathbb {R}^2 \curvearrowright (Z,\zeta )$ .

Since the action of $\{0\} \times \mathbb {R}$ scales the measure $\zeta '$ and $\mathbb {R} \curvearrowright Z'/(\{0\} \times \mathbb {R})$ is isomorphic to $\mathbb {R} \curvearrowright Y$ , we find a $\sigma $ -finite measure $\mu ' \sim \mu $ on Y and a measure-preserving isomorphism $\Delta : (Y \times \mathbb {R},\mu ' \times \gamma ) \to (Z',\zeta ')$ such that for all $(t,r) \in \mathbb {R}^2$ ,

$$ \begin{align*}\Delta(t \cdot y, \zeta(t,y) + r + s) = (t,r) \cdot \Delta(y,s) \quad\text{for a.e}.\ (y,s) \in Y \times \mathbb{R},\end{align*} $$

where $\zeta : \mathbb {R} \times Y \to \mathbb {R}$ is a $1$ -cocycle. Precomposing $\Delta $ with the measure-preserving map

$$ \begin{align*}(Y \times \mathbb{R},\mu \times \gamma) \to (Y \times \mathbb{R},\mu' \times \gamma) : (y,s) \mapsto (y, \log(d\mu' / d\mu)(y) + s)\end{align*} $$

and replacing $\zeta $ by a cohomologous $1$ -cocycle, we may assume that $\mu ' = \mu $ . Expressing that the action of $\mathbb {R} \times \{0\}$ scales the measure $\mu \times \gamma $ gives us that $\zeta (t,y) = \omega (t,y) + t$ . So we have found the required isomorphism.

Proof. Let $\mathbb {R}^2 \curvearrowright (Z,\zeta )$ be the unique action given by Proposition 3.1, associated with the ergodic flow $\mathbb {R} \curvearrowright (Y,\eta )$ . Consider the action

(3.2) $$ \begin{align} G \times \mathbb{R} \curvearrowright (X \times Z,\mu \times \zeta) : (g,r) \cdot (x,z) = (g\cdot x, (\omega(g,x),r) \cdot z). \end{align} $$

The action of $\{e\} \times \mathbb {R}$ scales the measure, while the action of $G \times \{0\}$ is measure-preserving. By definition, the action of G on $(X \times Z)/(\{e\} \times \mathbb {R})$ is isomorphic to $G \curvearrowright X \times Y$ . It thus follows that the action in (3.2) is the Maharam extension of $G \curvearrowright X \times Y$ together with its measure-scaling action of $\mathbb {R}$ .

By the uniqueness of $\mathbb {R}^2 \curvearrowright (Z,\zeta )$ , we may as well identify $(Z,\zeta ) = (\mathbb {R} \times \widehat {Y}, \gamma \times \widehat {\eta })$ with

$$ \begin{align*}(t,r) \cdot (s,\widehat{y}) = (t + r+\widehat{\beta}(r,\widehat{y}) + s , r \cdot \widehat{y}),\end{align*} $$

where $\widehat {\beta } : \mathbb {R} \times \widehat {Y} \to \mathbb {R}$ is the logarithm of the Radon–Nikodym cocycle for the adjoint flow $\mathbb {R} \curvearrowright ^{\widehat {\alpha }} \widehat {Y}$ . Then the action in (3.2) becomes

$$ \begin{align*}G \times \mathbb{R} &\curvearrowright (X \times \mathbb{R} \times \widehat{Y}, \mu \times \gamma \times \widehat{\eta}) : (g,r) \cdot (x,s,\widehat{y})\\ &= (g \cdot x, \omega(g,x) + \widehat{\beta}(r,\widehat{y}) + r + s, r \cdot \widehat{y}).\end{align*} $$

Since $G \curvearrowright (X,\mu )$ is ergodic and of type III $_1$ , the Maharam extension $G \curvearrowright X \times \mathbb {R}$ is ergodic. It follows that the G-invariant functions on $X \times \mathbb {R} \times \widehat {Y}$ are the functions that only depend on the $\widehat {Y}$ -variable. Since $\mathbb {R} \curvearrowright \widehat {Y}$ 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 $\widehat {\alpha } : \mathbb {R} \curvearrowright \widehat {Y}$ .

Proof. The implication from cocycle superrigidity to OE superrigidity was first proven, in a probability measure-preserving setting, in [Reference ZimmerZim84, Proposition 4.2.11]. The version that we need is literally proven in [Reference Drimbe and VaesDV21, Lemma 2.4].

Conversely, assume that $\Omega : G \times X \to \Lambda $ is a $1$ -cocycle with values in a countable group $\Lambda $ . Consider the free, non-singular, ergodic action

$$ \begin{align*}G \times \Lambda \curvearrowright X \times \Lambda : (g,a) \cdot (x,b) = (g \cdot x, \Omega(g,x) b a^{-1}).\end{align*} $$

By construction, this action is stably orbit equivalent to $G \curvearrowright X$ . Assume that this action is conjugate to an induction of $G \curvearrowright X$ . We have to prove that $\Omega $ is cohomologous to a group homomorphism.

Take an injective group homomorphism $\delta : G \to G \times \Lambda : \delta (g) = (\delta _1(g),\delta _2(g))$ and a measure space isomorphism $\Delta : X \to Z \subset X \times \Lambda $ such that $G \times \Lambda \curvearrowright X \times \Lambda $ is induced from $\delta (G) \curvearrowright Z$ and $\Delta $ is a conjugacy with respect to $\delta $ .

Since the action of $\Lambda $ on $X \times \Lambda $ admits a fundamental domain, the same is true for the action $\operatorname {Ker} \delta _1 \curvearrowright X$ . Since $G \curvearrowright X$ is simple, we find that $\delta _1$ is faithful. Since $\delta (G) \subset \delta _1(G) \times \Lambda $ and since $G \times \Lambda \curvearrowright X \times \Lambda $ is induced from $\delta (G) \curvearrowright Z$ , we find a fortiori that $G \times \Lambda \curvearrowright X \times \Lambda $ is induced from $\delta _1(G) \times \Lambda \curvearrowright Z_1$ with $Z_0 \subset Z_1$ . Since $Z_1$ is $\Lambda $ -invariant, we find that $Z_1 = X_0 \times \Lambda $ and conclude that $G \curvearrowright X$ is induced from $\delta _1(G) \curvearrowright X_0$ . Since $G \curvearrowright X$ is simple, we conclude that $\delta _1(G) = G$ . So $\delta _1$ is an automorphism of the group G.

Define the group homomorphism $\gamma : G \to \Lambda : \gamma = \delta _2 \circ \delta _1^{-1}$ . We have $\delta (g) = (\delta _1(g),\gamma (\delta _1(g)))$ and the map

$$ \begin{align*}\psi : (G \times \Lambda) / \delta(G) \to \Lambda : (g,k)\delta(G) \mapsto \gamma(g) k^{-1}\end{align*} $$

is a bijection satisfying $\psi ((g,a) \cdot i) = \gamma (g) \psi (i) a^{-1}$ for all $(g,a) \in G \times \Lambda $ and $i \in (G \times \Lambda )/\delta (G)$ .

Since $G \times \Lambda \curvearrowright X \times \Lambda $ is induced from $\delta (G) \curvearrowright Z$ , we find a $(G \times \Lambda )$ -equivariant map from $X \times \Lambda $ to $(G \times \Lambda ) / \delta (G)$ . We denote by $\theta $ its composition with $\psi $ . By $\Lambda $ -equivariance, we get that $\theta (x,a) = \theta _0(x) a$ , where $\theta _0 : X \to \Lambda $ is a Borel map. Expressing the G-equivariance gives us that

$$ \begin{align*}\Omega(g,x) = \theta_0(g \cdot x)^{-1} \, \gamma(g) \, \theta_0(x),\end{align*} $$

so that $\Omega $ is cohomologous to a group homomorphism.

Proof. Let $\Gamma _0 < \Gamma $ be a subgroup and $\pi : G \to \Gamma /\Gamma _0$ a Borel map such that $\pi (g h) = g \pi (h)$ for all $g \in \Gamma $ and a.e. $h \in G$ . We have to prove that $\pi $ is essentially constant. Since $\Gamma /\Gamma _0$ is countable, the set $\mathcal {U} = \{(h,k) \in G \times G \mid \pi (h) = \pi (k)\}$ is non-negligible.

Since $\pi (g h) = g \pi (h)$ when $g \in \Gamma $ , the set $\mathcal {U}$ is essentially invariant under the diagonal translation action $\Gamma \curvearrowright G \times G$ . By continuity, $\mathcal {U}$ is essentially invariant under the diagonal translation action of G. We thus find a non-negligible Borel set $\mathcal {V} \subset G$ such that, up to measure zero, $\mathcal {U} = \{(h,h k) \mid h \in G, k \in \mathcal {V}\}$ .

Define $G_0 = \{k \in G \mid \pi (h) = \pi (hk) \;\text {for a.e. }h \in G\}$ . Since $\mathcal {V}$ is non-negligible, $G_0$ is also non-negligible. By definition, $G_0$ is a subgroup of G. So $G_0$ must be an open subgroup of G. Since G is connected, we conclude that $G_0 = G$ . This means that $\pi $ is essentially constant.

Proof. Fix $g \in \mathcal {N}_\Gamma (\Gamma _0)$ and denote by $\alpha : \Gamma _0 \to \Gamma _0$ the automorphism $\alpha (h) = g h g^{-1}$ . Since for every $h \in \Gamma _0$ , the function $x \mapsto \Omega (h,x)$ factors through $\pi $ , we find a measurable family $(\delta _y)_{y \in Y}$ of group homomorphisms $\delta _y : \Gamma _0 \to \Lambda $ such that $\Omega (h,x) = \delta _{\pi (x)}(h)$ for all $h \in \Gamma _0$ and a.e. $x \in X$ . Applying the $1$ -cocycle relation to $g h = \alpha (h) g$ gives us that $\Omega (g,h \cdot x) \, \delta _{\pi (x)}(h) = \delta _{\pi (g \cdot x)}(\alpha (h)) \, \Omega (g,x)$ for a.e. $x \in X$ .

For every $g \in \mathcal {N}_\Gamma (\Gamma _0)$ , the map $x \mapsto \pi (g \cdot x)$ is $\Gamma _0$ -invariant. We can thus define the non-singular action $\mathcal {N}_\Gamma (\Gamma _0) \curvearrowright (Y,\eta )$ such that $\pi (g \cdot x) = g \cdot \pi (x)$ for all $g \in \mathcal {N}_\Gamma (\Gamma _0)$ and a.e. $x \in X$ . We conclude that for a.e. $y \in Y$ , we have

(4.1) $$ \begin{align} \Omega(g,h \cdot x) = \delta_{g \cdot y}(\alpha(h)) \, \Omega(g,x) \, \delta_y(h)^{-1} \quad\text{for }\mu_y\text{-a.e. }x \in X_y. \end{align} $$

The action $\Gamma _0 \curvearrowright (X_y,\mu _y)$ is ergodic and not induced. Also (4.1) is saying that the map $X_y \to \Lambda : x \mapsto \Omega (g,x)$ is $\Gamma _0$ -equivariant, where $\Gamma _0$ is acting on $\Lambda $ by $h \cdot \unicode{x3bb} = \delta _{g \cdot y}(\alpha (h)) \, \unicode{x3bb} \, \delta _y(h)^{-1}$ . By Lemma 2.1, the map $x \mapsto \Omega (g,x)$ is essentially constant on $(X_y,\mu _y)$ . Since this holds for a.e. $y \in Y$ , we have proven that $x \mapsto \Omega (g,x)$ factors through $\pi $ .

Proof of Theorem 4.4

By our assumption, the ring $\mathcal {A}$ contains a rational number $q \in \mathbb {Q} \setminus \mathbb {Z}$ or an irrational algebraic number $\alpha $ . In the first case, by taking a multiple of q, we find a prime p and a positive integer $N \in \mathbb {N} \setminus \{0\}$ with $p \nmid N$ such that $N p^{-1} \in \mathcal {A}$ . Since $p \nmid N$ , we can take integers $a,b \in \mathbb {Z}$ such that $aN + bp = 1$ , so that $p^{-1} = a Np^{-1} + b$ . We conclude that $p^{-1} \in \mathcal {A}$ and denote $\mathcal {A}_0 = \mathbb {Z}[p^{-1}]$ . In the second case, we define the algebraic number field $K = \mathbb {Q}(\alpha )$ . Let $d \geq 2$ be the degree of the minimal polynomial of $\alpha $ . The ring $\mathcal {O}_K$ of integers of K is a finitely generated $\mathbb {Z}$ -module that is contained in K, which has $\{1,\alpha ,\ldots ,\alpha ^{d-1}\}$ as a $\mathbb {Q}$ -vector space basis. We can thus take a positive integer $N \in \mathbb {N} \setminus \{0\}$ such that $N \mathcal {O}_K \subset \mathbb {Z}[\alpha ] \subset \mathcal {A}$ . We denote $\mathcal {A}_0 = \mathbb {Z} + N \mathcal {O}_K$ .

We start by proving that $E(n,\mathcal {A}_0) < \operatorname {SL}(n,\mathbb {R})$ is essentially cocycle superrigid with countable targets. When $\mathcal {A}_0 = \mathbb {Z}[p^{-1}]$ , we know by [Reference Hahn and O’MearaHOM89, Theorem 4.3.9] that $E(n,\mathcal {A}_0) = \operatorname {SL}(n,\mathcal {A}_0)$ and we know from [Reference Drimbe and VaesDV21, Proposition 4.1] that $\operatorname {SL}(n,\mathcal {A}_0) < \operatorname {SL}(n,\mathbb {R})$ is essentially cocycle superrigid with countable targets.

Next consider the case where $\mathcal {A}_0 = \mathbb {Z} + N \mathcal {O}_K$ . The ring $\mathcal {A}_1 = \mathcal {O}_K / (N \mathcal {O}_K)$ is finite and the kernel of the canonical homomorphism $\operatorname {SL}(n,\mathcal {O}_K) \to \operatorname {SL}(n,\mathcal {A}_1)$ is contained in $\operatorname {SL}(n,\mathcal {A}_0)$ . Thus, $\operatorname {SL}(n,\mathcal {A}_0) < \operatorname {SL}(n,\mathcal {O}_K)$ has finite index. Using the real and complex embeddings of K, the group $\operatorname {SL}(n,\mathcal {O}_K)$ is an irreducible lattice in a product of copies of $\operatorname {SL}(n,\mathbb {R})$ and $\operatorname {SL}(n,\mathbb {C})$ ; see, for example, [Reference Platonov and RapinchukPR94, Theorems 5.7 and 7.12]. The groups $\operatorname {SL}(n,\mathbb {R})$ and $\operatorname {SL}(n,\mathbb {C})$ both have property (T); see, for example, [Reference Bekka, de la Harpe and ValetteBHV08, Theorem 1.4.15]. Then the finite-index subgroup $\operatorname {SL}(n,\mathcal {A}_0)$ of $\operatorname {SL}(n,\mathcal {O}_K)$ is also such an irreducible lattice. In particular, $\operatorname {SL}(n,\mathcal {A}_0)$ satisfies Margulis’s normal subgroup theorem. Since $E(n,\mathcal {A}_0)$ is an infinite normal subgroup of $\operatorname {SL}(n,\mathcal {A}_0)$ , we conclude that $E(n,\mathcal {A}_0) < \operatorname {SL}(n,\mathcal {A}_0)$ has finite index. So $E(n,\mathcal {A}_0)$ is also an irreducible lattice in a product of copies of $\operatorname {SL}(n,\mathbb {R})$ and $\operatorname {SL}(n,\mathbb {C})$ . By part 2 of [Reference Drimbe and VaesDV21, Theorem C] (and this is essentially [Reference IoanaIoa14, Theorem B]), we get that $E(n,\mathcal {A}_0) < \operatorname {SL}(n,\mathbb {R})$ is essentially cocycle superrigid with countable targets.

We also need the following observation: given a countable dense subgroup $\Gamma $ of a connected Lie group G with universal cover $\pi : \widetilde {G} \to G$ , the subgroup $\Gamma < G$ is essentially cocycle superrigid with countable targets if and only if the translation action of the dense subgroup $\widetilde {\Gamma } = \pi ^{-1}(\Gamma )$ on $\widetilde {G}$ is cocycle superrigid with countable targets. One implication is obvious. So assume that $\Gamma < G$ is essentially cocycle superrigid with countable targets and that $\omega : \widetilde {\Gamma } \times \widetilde {G} \to \Lambda $ is a $1$ -cocycle with values in a countable group $\Lambda $ . Since the action of $\operatorname {Ker} \pi $ on $\widetilde {G}$ admits a fundamental domain, $\omega $ is cohomologous with a $1$ -cocycle $\omega _1$ satisfying $\omega _1(g,x) = e$ for all $g \in \operatorname {Ker} \pi $ and a.e. $x \in \widetilde {G}$ . This means that $\omega _1(g,x) = \omega _2(\pi (g),\pi (x))$ for all $g \in \widetilde {\Gamma }$ and a.e. $x \in \widetilde {G}$ , where $\omega _2 : \Gamma \times G \to \Lambda $ is a $1$ -cocycle. By our assumption, $\omega _1$ , and hence also $\omega $ , is cohomologous with a group homomorphism from $\widetilde {\Gamma }$ to $\Lambda $ .

We then prove statement (1). Take $E(n,\mathcal {A}) < \Gamma < \operatorname {SL}(n,\mathcal {A})$ . We write $G = \operatorname {SL}(n,\mathbb {R})$ and denote by $\pi : \widetilde {G} \to G$ its universal cover. Since $n \geq 3$ , $\operatorname {Ker} \pi $ is a central subgroup of order $2$ in $\widetilde {G}$ . Define $\widetilde {\Gamma } = \pi ^{-1}(\Gamma )$ . Let $\omega : \Gamma \times G \to \Lambda $ be a $1$ -cocycle, with lift $\widetilde {\omega } : \widetilde {\Gamma } \times \widetilde {G} \to \Lambda $ . We have to prove that $\widetilde {\omega }$ is cohomologous to a group homomorphism.

Since we have proven that $E(n,\mathcal {A}_0) < G$ is essentially cocycle superrigid, it follows from the observation above that $\widetilde {\omega }$ is cohomologous with a $1$ -cocycle $\gamma : \widetilde {\Gamma } \times \widetilde {G} \to \Lambda $ that has the property that $x \mapsto \gamma (g,x)$ is essentially constant for every $g \in \pi ^{-1}(E(n,\mathcal {A}_0))$ .

Write $[n] = \{1,\ldots ,n\}$ . For every $k \in [n]$ and for every subring $\mathcal {B} \subset \mathbb {R}$ , we define the following subgroups of $E(n,\mathcal {B})$ : the group $C_k(\mathcal {B}) \cong \mathcal {B}^{n-1}$ generated by $\{e_{ik}(b) \mid i \in [n] \setminus \{k\}, b \in \mathcal {B}\}$ ; the group $R_k(\mathcal {B}) \cong \mathcal {B}^{n-1}$ generated by $\{e_{kj}(b) \mid j \in [n] \setminus \{k\}, b \in \mathcal {B}\}$ ; and the group $H_k(\mathcal {B}) \cong E(n-1,\mathcal {B})$ generated by $\{e_{ij}(b) \mid i,j \in [n] \setminus \{k\}, i \neq j, b \in \mathcal {B}\}$ . Note that $H_k(\mathcal {B})$ normalizes both $C_k(\mathcal {B})$ and $R_k(\mathcal {B})$ . If $\mathcal {B} \subset \mathbb {R}$ is dense, $H_k(\mathcal {B})$ , $C_k(\mathcal {B})$ and $R_k(\mathcal {B})$ are also dense in $H_k(\mathbb {R})$ , $C_k(\mathbb {R})$ and $R_k(\mathbb {R})$ respectively, and the latter are closed subgroups of G.

Since $\mathbb {R}$ is simply connected, there is for all $i \neq j$ a unique continuous group homomorphism $\widetilde {e}_{ij} : \mathbb {R} \to \widetilde {G}$ such that $\pi (\widetilde {e}_{ij}(t)) = e_{ij}(t)$ for all $t \in \mathbb {R}$ . When $i,j \in [n] \setminus \{k\}$ and $s,t \in \mathbb {R}$ , the image $\pi ([\widetilde {e}_{ik}(t),\widetilde {e}_{jk}(s)])$ of the commutator equals the identity element. Thus, $[\widetilde {e}_{ik}(t),\widetilde {e}_{jk}(s)] \in \operatorname {Ker} \pi $ for all $s,t \in \mathbb {R}$ . By connectedness of $\mathbb {R}^2$ , we find that $\widetilde {e}_{ik}(t)$ commutes with $\widetilde {e}_{jk}(s)$ . There thus is a unique continuous group homomorphism $C_k(\mathbb {R}) \to \widetilde {G} : e_{ik}(t) \mapsto \widetilde {e}_{ik}(t)$ .

For every $k \in [n]$ and subring $\mathcal {B} \subset \mathbb {R}$ , we denote by $\widetilde {C}_k(\mathcal {B})$ the subgroup of $\widetilde {G}$ generated by $\{\widetilde {e}_{ik}(b) \mid i \in [n] \setminus \{k\}, b \in \mathcal {B}\}$ . Note that $\widetilde {C}_k(\mathbb {R})$ is a connected closed subgroup of $\widetilde {G}$ and that $\pi : \widetilde {C}_k(\mathbb {R}) \to C_k(\mathbb {R})$ is an isomorphism. We also have that $\widetilde {C}_k(\mathbb {R})$ is the connected component of the identity in $\pi ^{-1}(C_k(\mathbb {R}))$ and that $\widetilde {C}_k(\mathcal {B}) = \widetilde {C}_k(\mathbb {R}) \cap \pi ^{-1}(C_k(\mathcal {B}))$ .

Define $c_k : \widetilde {G} \to \widetilde {C}_k(\mathbb {R}) \backslash \widetilde {G} : c_k(x) = \widetilde {C}_k(\mathbb {R}) x$ . Since $\widetilde {C}_k(\mathcal {A})$ commutes with $\widetilde {C}_k(\mathcal {A}_0)$ and since $\widetilde {C}_k(\mathcal {A}_0)$ is dense in $\widetilde {C}_k(\mathbb {R})$ , it follows from Lemmas 4.5 and 4.6 that for every $g \in \widetilde {C}_k(\mathcal {A})$ , the map $x \mapsto \gamma (g,x)$ factors through $c_k$ .

Take $g \in \pi ^{-1}(H_k(\mathcal {A}) C_k(\mathcal {A}))$ . Since g normalizes $\pi ^{-1}(C_k(\mathbb {R}))$ , the element g also normalizes its connected component of the identity $\widetilde {C}_k(\mathbb {R})$ . Since g normalizes $\pi ^{-1}(C_k(\mathcal {A}))$ as well, it follows that g normalizes $\widetilde {C}_k(\mathcal {A})$ . Another application of Lemmas 4.5 and 4.6 then says that the map $x \mapsto \gamma (g,x)$ factors through $c_k$ .

Fix $i \neq j$ and fix $g \in \pi ^{-1}(e_{ij}(\mathcal {A}))$ . In the following paragraphs, we prove that $x \mapsto \gamma (g,x)$ is essentially constant.

We have proven that the map $x \mapsto \gamma (g,x)$ factors through $c_k$ for all $k \neq i$ . This means that for all $b \neq i$ , $a \neq b$ and $t \in \mathbb {R}$ , we have $\gamma (g,\widetilde {e}_{ab}(t)x) = \gamma (g,x)$ for a.e. $x \in \widetilde {G}$ . Since we can reason analogously using the subgroups $R_k$ , we also get for all $a \neq j$ , $b \neq a$ and $t \in \mathbb {R}$ that $\gamma (g,\widetilde {e}_{ab}(t)x) = \gamma (g,x)$ for a.e. $x \in \widetilde {G}$ .

Define $\widetilde {T} = \{h \in \widetilde {G} \mid \gamma (g,hx) = \gamma (g,x) \;\;\text {for a.e. }x \in \widetilde {G}\;\}$ . Then $\widetilde {T}$ is a closed subgroup of $\widetilde {G}$ . Define $T = \pi (\widetilde {T})$ . By the previous paragraph, $e_{ab}(\mathbb {R}) \subset T$ when $a \neq b$ , $a \neq j$ or $b \neq i$ . When $a = j$ and $b =i$ , we choose $c \in [n] \setminus \{i,j\}$ and note that for all $t \in \mathbb {R}$ ,

$$ \begin{align*}e_{ac}(t) \, e_{cb}(1) \, e_{ac}(-t) \, e_{cb}(-1) = e_{ab}(t),\end{align*} $$

so that again $e_{ab}(\mathbb {R}) \subset T$ . It follows that $T = G$ . The closed subgroup $\widetilde {T} \subset \widetilde {G}$ thus has index at most $2$ , so that $\widetilde {T} \subset \widetilde {G}$ is open. Since $\widetilde {G}$ is connected, it follows that $\widetilde {T} = \widetilde {G}$ . This means that $x \mapsto \gamma (g,x)$ is essentially constant.

We have thus proven that $x \mapsto \gamma (g,x)$ is essentially constant for every $g \in \pi ^{-1}(E(n,\mathcal {A}))$ . Since $\pi ^{-1}(E(n,\mathcal {A}))$ is a normal subgroup of $\widetilde {\Gamma }$ , a final application of Lemmas 4.5 and 4.6 implies that $x \mapsto \gamma (g,x)$ is essentially constant for every $g \in \widetilde {\Gamma }$ . This concludes the proof of statement (1).

To prove statement (2), we still write $G = \operatorname {SL}(n,\mathbb {R})$ with universal cover $\pi : \widetilde {G} \to G$ . Note that $\widetilde {G} \ltimes \mathbb {R}^n$ is the universal cover of $G \ltimes \mathbb {R}^n$ . Let $\omega : (\Gamma \ltimes \mathcal {A}^n) \times (G \ltimes \mathbb {R}^n) \to \Lambda $ be a $1$ -cocycle, with lift $\widetilde {\omega } : (\widetilde {\Gamma } \ltimes \mathcal {A}^n) \times (\widetilde {G} \ltimes \mathbb {R}^n) \to \Lambda $ .

By the same argument as at the beginning of this proof, $E(n,\mathcal {A}_0) \ltimes \mathcal {A}_0^n$ is an irreducible lattice in a product of copies of $\operatorname {SL}(n,\mathbb {R}) \ltimes \mathbb {R}^n$ and $\operatorname {SL}(n,\mathbb {C}) \ltimes \mathbb {C}^n$ . By [Reference Bekka, de la Harpe and ValetteBHV08, Corollary 1.4.16], these groups have property (T). By part 2 of [Reference Drimbe and VaesDV21, Theorem C], it follows that the dense subgroup $E(n,\mathcal {A}_0) \ltimes \mathcal {A}_0^n$ of $\operatorname {SL}(n,\mathbb {R}) \ltimes \mathbb {R}^n$ is essentially cocycle superrigid with countable targets.

By the observation at the beginning of the proof, we thus find that $\widetilde {\omega }$ is cohomologous with a $1$ -cocycle $\gamma $ such that $x \mapsto \gamma (g,x)$ is essentially constant for every $g \in \pi ^{-1}(E(n,\mathcal {A}_0)) \ltimes \mathcal {A}_0^n$ . Denote by $\theta : \widetilde {G} \ltimes \mathbb {R}^n \to \widetilde {G}$ the natural quotient map. Since $\mathcal {A}_0^n$ is dense in $\mathbb {R}^n$ and since $\mathcal {A}^n$ commutes with $\mathcal {A}_0^n$ , it follows from Lemmas 4.5 and 4.6 that $x \mapsto \gamma (a,x)$ factors through $\theta $ for all $a \in \mathcal {A}^n$ . Since $\mathcal {A}^n$ is a normal subgroup of $\widetilde {\Gamma } \ltimes \mathcal {A}^n$ , another application of Lemmas 4.5 and 4.6 implies that $x \mapsto \gamma (g,x)$ factors through $\theta $ for all $g \in \widetilde {\Gamma } \ltimes \mathcal {A}^n$ .

We thus find a Borel map $\gamma _1 : (\widetilde {\Gamma } \ltimes \mathcal {A}^n) \times \widetilde {G} \to \Lambda $ such that for all $g \in \widetilde {\Gamma } \ltimes \mathcal {A}^n$ and a.e. $x \in \widetilde {G} \ltimes \mathbb {R}^n$ , we have $\gamma (g,x) = \gamma _1(g,\theta (x))$ . From now on, we denote by g the elements of $\widetilde {\Gamma }$ and we denote by a the elements of $\mathcal {A}^n$ . The restriction of $\gamma _1$ to $\widetilde {\Gamma } \times \widetilde {G}$ is a $1$ -cocycle for the translation action $\widetilde {\Gamma } \curvearrowright \widetilde {G}$ . Above, we have proven that this action is cocycle superrigid with countable target groups. Choose a Borel map $\varphi : \widetilde {G} \to \Lambda $ and a group homomorphism $\delta : \widetilde {\Gamma } \to \Lambda $ such that $\gamma _1(g,y) = \varphi (gy)^{-1} \, \delta (g) \, \varphi (y)$ for all $g \in \widetilde {\Gamma }$ and a.e. $y \in \widetilde {G}$ . Replacing $\gamma $ with the cohomologous $1$ -cocycle $(g,x) \mapsto \varphi (\theta (gx)) \, \gamma (g,x) \, \varphi (\theta (x))^{-1}$ , we may thus assume that $\gamma _1(g,y) = \delta (g)$ for all $g \in \widetilde {\Gamma }$ and a.e. $y \in \widetilde {G}$ . We denote $\delta _y(a) = \gamma _1(a,y)$ and note that $\delta _y : \mathcal {A}^n \to \Lambda $ is a measurable family of group homomorphisms. To conclude the proof of the theorem, we have to show that for all $a \in \mathcal {A}^n$ , the map $y \mapsto \delta _y(a)$ is essentially constant.

When $g \in \widetilde {\Gamma }$ , we have that $\pi (g) \in \operatorname {SL}(n,\mathcal {A}^n)$ so that $\pi (g)(a) \in \mathcal {A}^n$ . The group law in $\widetilde {\Gamma } \ltimes \mathcal {A}^n$ can then be expressed by $g \, a = \pi (g)(a) \, g$ for all $g \in \widetilde {\Gamma }$ , $a \in \mathcal {A}^n$ . Applying the $1$ -cocycle relation for $\gamma $ , we conclude that

(4.2) $$ \begin{align} \delta_{gy}(\pi(g)a) = \delta(g) \, \delta_y(a) \, \delta(g)^{-1} \quad\text{for all }g \in \widetilde{\Gamma}, a \in \mathcal{A}^n\text{ and a.e. }y \in \widetilde{G}. \end{align} $$

Fix $i \in [n]$ . Using the notation introduced above, denote by $L_i(\mathcal {A})$ the subgroup of $\Gamma $ generated by $H_i(\mathcal {A})$ and $R_i(\mathcal {A})$ . Similarly define $L_i(\mathbb {R})$ . Then $L_i(\mathcal {A})$ is dense in $L_i(\mathbb {R})$ and $L_i(\mathbb {R})$ consists of the matrices $A \in \operatorname {SL}(n,\mathbb {R})$ satisfying $A(e_i) = e_i$ , where $e_i$ is the ith standard basis vector. Since the inclusion $\operatorname {SL}(n-1,\mathbb {R}) \cong H_i(\mathbb {R}) < \operatorname {SL}(n,\mathbb {R})$ induces a surjective homomorphism between the fundamental groups, we get that $\pi ^{-1}(L_i(\mathbb {R}))$ is a connected subgroup of $\widetilde {G}$ .

For every $a \in \mathcal {A}$ , denote $e_i(a) = a e_i \in \mathcal {A}^n$ . It follows from (4.2) that for all $g \in \pi ^{-1}(L_i(\mathcal {A}))$ , we have $\delta _{gy}(e_i(a)) = \delta (g) \, \delta _y(e_i(a)) \, \delta (g)^{-1}$ . By Lemma 4.5, for every fixed $i \in [n]$ and $a \in \mathcal {A}$ , the map $y \mapsto \delta _y(e_i(a))$ is invariant under left translation by the connected group $\pi ^{-1}(L_i(\mathbb {R}))$ , and thus of the form $y \mapsto \zeta (\pi (y)^{-1}(e_i))$ for some Borel map $\zeta : \mathbb {R}^n \to \Lambda $ . This means that we find measurable families $(\rho _{i,z})_{z \in \mathbb {R}^n}$ of group homomorphisms $\rho _{i,z} : \mathcal {A} \to \Lambda $ such that $\delta _y(e_i(a)) = \rho _{i,\pi (y)^{-1}(e_i)}(a)$ for all $i \in [n]$ , $a \in \mathcal {A}$ and a.e. $y \in \widetilde {G}$ .

Take $i \neq j$ . We now apply (4.2) for $g \in \widetilde {\Gamma }$ with $\pi (g) = e_{ij}(-1)$ and $e_j(a) \in \mathcal {A}^n$ . We conclude that

$$ \begin{align*}\delta_{gy}(e_j(a)) \, \delta_{gy}(e_i(a))^{-1} &= \delta_{gy}(e_j(a) - e_i(a)) = \delta_{gy}(\pi(g)e_j(a))\\ &= \delta(g) \, \delta_y(e_j(a)) \, \delta(g)^{-1}.\end{align*} $$

Since $\pi (g)^{-1}(e_i) = e_i$ and $\pi (g)^{-1}(e_j) = e_i + e_j$ , we find that

$$ \begin{align*}\rho_{j,\pi(y)^{-1}(e_i) + \pi(y)^{-1}(e_j)}(a) \, \rho_{i,\pi(y)^{-1}(e_i)}(a)^{-1} = \delta(g) \, \rho_{j,\pi(y)^{-1}(e_j)}(a) \, \delta(g)^{-1}\end{align*} $$

for all $a \in \mathcal {A}$ and a.e. $y \in \widetilde {G}$ . Since $n \geq 3$ , the map $\widetilde {G} \to \mathbb {R}^n \times \mathbb {R}^n : y \mapsto (\pi (y)^{-1}(e_i), \pi (y)^{-1}(e_j))$ is a non-singular factor map. We thus conclude that

(4.3) $$ \begin{align} \rho_{j,u+v}(a) \, \rho_{i,u}(a)^{-1} = \delta(g) \, \rho_{j,v}(a) \, \delta(g)^{-1} \quad\text{for all }a \in \mathcal{A}\text{ and a.e. }(u,v) \in \mathbb{R}^n \times \mathbb{R}^n. \end{align} $$

Denote by $\mathcal {P}$ the Polish group of Borel maps from $\mathbb {R}^n$ to $\Lambda $ , 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 $v \in \mathbb {R}^n$ and $F \in \mathcal {P}$ , define $F_v \in \mathcal {P}$ by $F_v(u) = F(u+v)$ . Then the map $\mathbb {R}^n \to \mathcal {P} : v \mapsto F_v$ is continuous. Fix $a \in \mathcal {A}$ and define $F,G \in \mathcal {P}$ by $F(u) = \rho _{j,u}(a)$ and $G(u) = \rho _{i,u}(a)^{-1}$ . Then the map $\mathbb {R}^n \to \mathcal {P} : v \mapsto F_v \, G$ is continuous. By (4.3), this map takes values in the discrete subgroup $\Lambda < \mathcal {P}$ of constant functions. Since $\mathbb {R}^n$ is connected, it follows that $v \mapsto F_v$ is essentially constant. That means that we find group homomorphisms $\rho _j : \mathcal {A} \to \Lambda $ such that $\rho _{j,u}(a) = \rho _j(a)$ for a.e. $y \in \mathbb {R}^n$ .

We conclude that for all $j \in [n]$ and every $a \in \mathcal {A}$ , the map $y \mapsto \delta _y(e_j(a))$ is essentially constant. So $y \mapsto \delta _y(a)$ is also essentially constant for every $a \in \mathcal {A}^n$ . This concludes the proof of the theorem.

Proof. Since $\mathcal {A}^n$ is dense in $\mathbb {R}^n$ , by Lemma 4.5, the action $\mathcal {A}^n \curvearrowright \mathbb {R}^n$ is ergodic and not induced. A fortiori, $G \curvearrowright \mathbb {R}^n$ is ergodic and not induced. Assume that $\Sigma \lhd G$ is a normal subgroup whose action on $\mathbb {R}^n$ admits a fundamental domain. Write $\Sigma _0 = \Sigma \cap \mathcal {A}^n$ . Let $(A,a) \in \Sigma $ . If $A \neq 1$ , then for all $b \in \mathcal {A}^n$ ,

$$ \begin{align*}(1,(1-A)b) = (1,b) (A,a) (1,b)^{-1} (A,a)^{-1} \in \Sigma\end{align*} $$

so that $\Sigma _0 \neq \{0\}$ . Since $\Sigma _0$ is globally invariant under $\operatorname {SL}(n,\mathcal {A})$ , the closure of $\Sigma _0$ in $\mathbb {R}^n$ is a non-trivial closed subgroup of $\mathbb {R}^n$ that is globally invariant under $\operatorname {SL}(n,\mathbb {R})$ . So $\Sigma _0$ is dense in $\mathbb {R}^n$ . It follows that $\Sigma _0 \curvearrowright \mathbb {R}^n$ is ergodic, contradicting the assumption that $\Sigma _0 \curvearrowright \mathbb {R}^n$ admits a fundamental domain. So $A = 1$ , and we have proven that $\Sigma \subset \mathcal {A}^n$ . If $\Sigma \neq \{0\}$ , we again find that $\Sigma $ is dense in $\mathbb {R}^n$ . So $\Sigma $ is trivial, and we have proven that $G \curvearrowright \mathbb {R}^n$ is a simple action.

Write $G_0 = \operatorname {SL}(n,\mathcal {A}) \ltimes \mathcal {A}^n$ . By Theorem 4.4, the dense subgroup $G_0$ of $\operatorname {SL}(n,\mathbb {R}) \ltimes \mathbb {R}^n$ is essentially cocycle superrigid with countable targets. By [Reference Drimbe and VaesDV21, Proposition 3.3], the action $G_0 \curvearrowright \mathbb {R}^n$ 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 $G \curvearrowright \mathbb {R}^n$ is also cocycle superrigid with countable targets. By Proposition 4.2, the action is also OE superrigid (v1).

The Maharam extension of $G \curvearrowright \mathbb {R}^n$ can be identified with the action $G \curvearrowright \mathbb {R}^n \times \mathbb {R}$ given by

$$ \begin{align*}(A,a) \cdot (x,s) = (A(a+x),\log |\det A| + s).\end{align*} $$

Since the translation action $\mathcal {A}^n \curvearrowright \mathbb {R}^n$ is ergodic, it follows that the G-invariant functions on $\mathbb {R}^n \times \mathbb {R}$ are precisely the functions on $\mathbb {R}$ that are invariant under translation by all $\log |\det A|$ , $A \in \operatorname {GL}(n,\mathcal {A})$ , $\det A \in \mathcal {F}$ . So these are the functions on $\mathbb {R}$ that are invariant under translation by $\{\log |a| \mid a \in \mathcal {F}\}$ , so that the type of $G \curvearrowright \mathbb {R}^n$ is as described in the corollary.

Proof. Let $G \curvearrowright (X,\mu )$ be a free, ergodic, non-singular action of type III $_0$ . Assume that every $1$ -cocycle $\Omega : G \times X \to \mathbb {Z}$ is cohomologous to a group homomorphism. We prove that $G \curvearrowright X$ must be an induced action. By Proposition 4.2, this suffices to prove the proposition.

Combining [Reference SchmidtSch79, Theorem 2.7 and Remark 2.9] and [Reference Jones and SchmidtJS85, Theorem 2.1], we find a free, ergodic, probability measure-preserving action $\mathbb {Z} \curvearrowright (Y,\eta )$ and a Borel map $\pi : X \to Y$ such that $\pi _*(\mu ) \sim \eta $ and $\pi (G \cdot x) \subset \mathbb {Z} \cdot \pi (x)$ for a.e. $x \in X$ . Since all free, ergodic, probability measure-preserving actions of $\mathbb {Z}$ are orbit equivalent, we may assume that the action $\mathbb {Z} \curvearrowright (Y,\eta )$ is the profinite $\mathbb {Z} \curvearrowright \mathbb {Z}_2$ , viewed as the inverse limit of $\mathbb {Z} \curvearrowright \mathbb {Z} / 2^k \mathbb {Z}$ , $k \in \mathbb {N}$ .

Define the $1$ -cocycle $\Omega : G \times X \to \mathbb {Z}$ such that $\pi (g \cdot x) = \Omega (g,x) \cdot \pi (x)$ for all $g \in G$ and a.e. $x \in X$ . By our assumption, we find a Borel map $\varphi : X \to \mathbb {Z}$ and a group homomorphism $\delta : G \to \mathbb {Z}$ such that $\Omega (g,x) = -\varphi (g \cdot x) + \delta (g) + \varphi (x)$ for all $g \in G$ and a.e. $x \in X$ . Define the Borel map $\pi _1 : X \to Y : \pi _1(x) = \varphi (x) \cdot \pi (x)$ . By construction, $\pi _1(g \cdot x) = \delta (g) \cdot \pi _1(x)$ .

Note that $\pi _1$ is not essentially constant, since otherwise $\pi (x)$ takes values in a countable set for a.e. $x \in X$ , contradicting $\pi _*(\mu ) \sim \eta $ . We can then choose $k \in \mathbb {N}$ large enough such that, denoting by $\psi : \mathbb {Z}_2 \to \mathbb {Z} / 2^k \mathbb {Z}$ the canonical quotient map, $\theta = \psi \circ \pi $ is not essentially constant. Since $\theta (g \cdot x) = \delta (g) + \theta (x)$ for all $g \in G$ and a.e. $x \in X$ , it follows that the action $G \curvearrowright (X,\mu )$ is induced in a non-trivial way.

Proof of Theorem C

This now follows immediately from Corollary 4.7 and Proposition 4.8.

Proof. Define the $1$ -cocycle $\widetilde {\Omega } : (G \times \mathbb {R}) \times \widetilde {X} \to \Lambda : \widetilde {\Omega }((g,t),(x,s)) = \Omega (g,x)$ . First restrict $\widetilde {\Omega }$ to a $1$ -cocycle for the action $G \curvearrowright \widetilde {X}$ with ergodic decomposition given by $\pi : \widetilde {X} \to Y$ . As explained in detail in [Reference Fisher, Morris and WhyteFMW04], we may consider $\widetilde {\Omega }$ as a measurable family $(\widetilde {\Omega }_y)_{y \in Y}$ of $1$ -cocycles for the measurable family of actions $G \curvearrowright (\widetilde {X}_y,\widetilde {\mu }_y)$ . By assumption, $\eta $ -a.e. $\widetilde {\Omega }_y$ is cohomologous to a group homomorphism $\delta _y : G \to \Lambda $ .

By [Reference Fisher, Morris and WhyteFMW04, Corollary 3.11], we find a Borel family of group homomorphisms $\delta _y : G \to \Lambda $ , indexed by $y \in Y$ , and a Borel map $\varphi : \widetilde {X} \to \Lambda $ such that for all $g \in G$ , we have that

$$ \begin{align*}\widetilde{\Omega}((g,0),(x,s)) = \varphi((g,0) \cdot (x,s))^{-1} \, \delta_{\pi(x,s)}(g) \, \varphi(x,s) \quad\text{for }\widetilde{\mu}\text{-a.e. }(x,s) \in \widetilde{X}.\end{align*} $$

Define the $1$ -cocycle $\Psi : (G \times \mathbb {R}) \times \widetilde {X} \to \Lambda $ by

$$ \begin{align*}\Psi((g,t),(x,s)) = \varphi((g,t) \cdot (x,s)) \, \widetilde{\Omega}((g,t),(x,s)) \, \varphi(x,s)^{-1}.\end{align*} $$

By construction, $\Psi \sim \widetilde {\Omega }$ as $1$ -cocycles for $G \times \mathbb {R} \curvearrowright \widetilde {X}$ and for all $g \in G$ , $\Psi ((g,0),(x,s)) = \delta _{\pi (x,s)}(g)$ for $\widetilde {\mu }$ -a.e. $(x,s) \in \widetilde {X}$ . Define $\zeta _t(x,s) = \Psi ((e,t),(x,s))$ . From the $1$ -cocycle relation for $\Psi $ applied to $(g,0)(e,t) = (g,t) = (e,t)(g,0)$ , it follows that for all $t \in \mathbb {R}$ , $g \in G$ , we have

(5.1) $$ \begin{align} \zeta_t(g \cdot (x,s)) = \delta_{t \cdot \pi(x,s)}(g) \, \zeta_t(x,s) \, \delta_{\pi(x,s)}(g)^{-1} \quad\text{for }\widetilde{\mu}\text{-a.e. }(x,s) \in \widetilde{X}. \end{align} $$

Fix $t \in \mathbb {R}$ . Then (5.1) is saying that for $\eta $ -a.e. $y \in Y$ , $\zeta _t$ is a G-equivariant Borel map from $(\widetilde {X}_y,\widetilde {\mu }_y)$ to the countable set $\Lambda $ on which G is acting by $g \cdot \unicode{x3bb} = \delta _{t \cdot y}(g) \unicode{x3bb} \delta _y(g)^{-1}$ . Since we assumed that for $\eta $ -a.e. $y \in Y$ , the action $G \curvearrowright (\widetilde {X}_y,\widetilde {\mu }_y)$ is not induced, it follows that for $\eta $ -a.e. $y \in Y$ , the map $\zeta _t$ is $\widetilde {\mu }_y$ -almost everywhere constant on $\widetilde {X}_y$ .

We thus find a Borel map $\gamma _0 : \mathbb {R} \times Y \to \Lambda $ such that for all $t \in \mathbb {R}$ , we have that $\zeta _t(x,s) = \gamma _0(t,\pi (x,s))$ for $\widetilde {\mu }$ -a.e. $(x,s) \in \widetilde {X}$ . Since $\widetilde {\Omega }((e,t),(x,s)) = e$ , we have $\zeta _t(x,s) = \varphi (x,t+s) \, \varphi (x,s)^{-1}$ for $\widetilde {\mu }$ -a.e. $(x,s) \in \widetilde {X}$ . Therefore, for every $t \in \mathbb {R}$ ,

(5.2) $$ \begin{align} \gamma_0(t,\pi(x,s)) = \varphi(x,t+s) \, \varphi(x,s)^{-1} \quad\text{for }\widetilde{\mu}\text{-a.e. }(x,s) \in \widetilde{X}. \end{align} $$

So $\gamma _0 : \mathbb {R} \times Y \to \Lambda $ is a $1$ -cocycle. Then (5.1) is saying that for all $t \in \mathbb {R}$ , $g \in G$ , we have

(5.3) $$ \begin{align} \delta_{t \cdot y}(g) = \gamma_0(t,y) \, \delta_y(g) \, \gamma_0(t,y)^{-1} \quad\text{for }\eta\text{-a.e. }y \in Y. \end{align} $$

Since G is finitely generated, the set of group homomorphisms from G to $\Lambda $ is countable. We thus find a group homomorphism $\delta : G \to \Lambda $ such that $\delta _y = \delta $ for all y in a non-negligible Borel subset $\mathcal {U} \subset Y$ . Combining (5.3) with the ergodicity of $\mathbb {R} \curvearrowright (Y,\eta )$ , it follows that $\delta _y$ is conjugate to $\delta $ for $\eta $ -a.e. $y \in Y$ . We then find a Borel map $\rho : Y \to \Lambda $ so that $\delta _y(g) = \rho (y)^{-1} \, \delta (g) \, \rho (y)$ for $\eta $ -a.e. $y \in Y$ and all $g \in G$ . Replacing $\varphi (x,s)$ by $\rho (\pi (x,s)) \varphi (x,s)$ , we may assume that $\delta _y = \delta $ for $\eta $ -a.e. $y \in Y$ .

The $1$ -cocycle $\Psi : (G \times \mathbb {R}) \times \widetilde {X} \to \Lambda $ thus has the property that for all $g \in G$ and $t \in \mathbb {R}$ ,

$$ \begin{align*}\Psi((g,0),(x,s)) = \delta(g) \quad\text{and}\quad \Psi((e,t),(x,s)) = \gamma_0(t,\pi(x,s))\end{align*} $$

for $\widetilde {\mu }$ -a.e. $(x,s) \in \widetilde {X}$ . The cocycle identity for $\Psi $ then forces $\gamma _0$ to take values almost everywhere in the centralizer $C_\Lambda (\delta (G))$ .

Choose a strict $1$ -cocycle $\gamma : \mathbb {R} \times Y \to C_\Lambda (\delta (G))$ such that for every $t \in \mathbb {R}$ , we have that $\gamma (t,y) = \gamma _0(t,y)$ for $\eta $ -a.e. $y \in Y$ . Define the Borel map $\theta : \widetilde {X} \to C_\Lambda (\delta (G)) : \theta (x,s) = \gamma (s,\psi (x))$ . Consider the cohomologous $1$ -cocycle $\Psi _1 \sim \Psi $ defined by

$$ \begin{align*}\Psi_1((g,t),(x,s)) = \theta((g,t)\cdot(x,s))^{-1} \, \Psi((g,t),(x,s)) \, \theta(x,s).\end{align*} $$

Since $\gamma $ is a strict $1$ -cocycle, we find for every $t \in \mathbb {R}$ and $g \in G$ that $\Psi _1((e,t),(x,s)) = e$ and $\Psi _1((g,0),(x,s)) = \delta (g) \, \gamma (-\omega (g,x),\psi (x))$ for $\widetilde {\mu }$ -a.e. $(x,s) \in \widetilde {X}$ . So, defining the $1$ -cocycle

$$ \begin{align*}\Psi_0 : G \times X \to \Lambda : \Psi_0(g,x) = \delta(g) \, \gamma(-\omega(g,x),\psi(x)),\end{align*} $$

we have proven that the $1$ -cocycles $\Omega $ and $\Psi _0$ are cohomologous when viewed as $1$ -cocycles for $G \times \mathbb {R} \curvearrowright \widetilde {X}$ . The $\mathbb {R}$ -invariance of both $1$ -cocycles forces the Borel function implementing the cohomology $\Omega \sim \Psi _0$ to be essentially $\mathbb {R}$ -invariant as well. We have thus proven that $\Omega \sim \Psi _0$ .

Proof. As in the proof of Proposition 3.4, we take the unique action $\mathbb {R}^2 \curvearrowright (Z,\zeta )$ given by Proposition 3.1, associated with the ergodic flow $\mathbb {R} \curvearrowright (Y,\eta )$ . Identify $Y = Z / (\{0\} \times \mathbb {R})$ and denote by $\pi _1 : Z \to Y$ the corresponding factor map. Also write $\widehat {Y} = Z/(\mathbb {R} \times \{0\})$ and denote by $\pi _2 : Z \to \widehat {Y}$ the corresponding factor map. The Maharam extension of $G \curvearrowright X \times Y$ together with its measure-scaling action of $\mathbb {R}$ is then given by

$$ \begin{align*}G \times \mathbb{R} \curvearrowright X \times Z : (g,t) \cdot (x,z) = (g\cdot x, (\omega(g,x),t) \cdot z).\end{align*} $$

The map $(x,z) \mapsto \pi _2(z)$ identifies the associated flow of the action $G \curvearrowright X \times Y$ with $\mathbb {R} \curvearrowright \widehat {Y}$ . Identifying $Z = \mathbb {R} \times \widehat {Y}$ , the ergodic decomposition of $G \curvearrowright X \times Z$ is almost everywhere given by the Maharam extension $G \curvearrowright X \times \mathbb {R}$ of the initial type III $_1$ action $G \curvearrowright X$ . So $G \curvearrowright X \times Y$ satisfies the assumptions of Theorem 5.1.

Let $\Lambda $ be a countable group and $\Omega : G \times X \times Y \to \Lambda $ a $1$ -cocycle. Define the $1$ -cocycle

$$ \begin{align*}\widetilde{\Omega} : (G \times \mathbb{R}) \times (X \times Z) \to \Lambda : \widetilde{\Omega}((g,t),(x,z)) = \Omega(g,(x,\pi_1(z))).\end{align*} $$

It follows from Theorem 5.1 and Remark 5.2 that $\widetilde {\Omega }$ is cohomologous with $\Omega _1$ , where

$$ \begin{align*}\Omega_1((g,t),(x,z)) = \delta(g) \, \gamma_1(t,\pi_2(z)),\end{align*} $$

with $\delta : G \to \Lambda $ a group homomorphism and $\gamma _1 : \mathbb {R} \times \widehat {Y} \to C_\Lambda (\delta (G))$ a $1$ -cocycle. Define the $1$ -cocycle

$$ \begin{align*}\widetilde{\gamma}_1 : \mathbb{R}^2 \times Z \to C_\Lambda(\delta(G)) : \widetilde{\gamma}_1((r,t),z) = \gamma_1(t,\pi_2(z)).\end{align*} $$

Since the action of $\{0\} \times \mathbb {R}$ on Z is measure-scaling, $\widetilde {\gamma }_1$ is cohomologous to a $1$ -cocycle $\gamma _2$ of the form $\gamma _2((r,t),z) = \gamma (r,\pi _1(z))$ , where $\gamma : \mathbb {R} \times Y \to C_\Lambda (\delta (G))$ is a $1$ -cocycle. Choose a Borel map $\varphi : Z \to C_\Lambda (\delta (G))$ implementing this cohomology, so that for all $(r,t) \in \mathbb {R}^2$ , we have

$$ \begin{align*}\gamma(r,\pi_1(z)) = \varphi((r,t) \cdot z) \, \gamma_1(t,\pi_2(z)) \, \varphi(z)^{-1} \quad\text{for a.e. }z \in Z.\end{align*} $$

Define the $1$ -cocycle $\Omega _2 : (G \times \mathbb {R}) \times (X \times Z) \to \Lambda $ by

$$ \begin{align*}\Omega_2((g,t),(x,z)) = \varphi((\omega(g,x),t) \cdot z) \, \Omega_1((g,t),(x,z)) \, \varphi(z)^{-1}.\end{align*} $$

By construction, $\widetilde {\Omega } \sim \Omega _2$ and

$$ \begin{align*}\Omega_2((g,t),(x,z)) = \delta(g) \, \gamma(\omega(g,x),\pi_1(z)).\end{align*} $$

Since both $\widetilde {\Omega }$ and $\Omega _2$ are trivial on $\{0\} \times \mathbb {R}$ , this means that $\Omega $ is cohomologous with the $1$ -cocycle

$$ \begin{align*}G \times X \times Y : (g,x,y) \mapsto \delta(g) \, \gamma(\omega(g,x),y).\\[-37pt] \end{align*} $$

Proof. Let $\Gamma \curvearrowright (Z,\zeta )$ be a free, ergodic, non-singular action and let $\Delta : \mathcal {U} \subset X \to Z$ be a stable orbit equivalence between $G \curvearrowright X$ and $\Gamma \curvearrowright Z$ . By ergodicity of $G \curvearrowright (X,\mu )$ , we can choose a Borel map $\theta : X \to G$ such that $\theta (x) = e$ for all $x \in \mathcal {U}$ and $\theta (x) \cdot x \in \mathcal {U}$ for a.e. $x \in X$ . Define $\Delta _0 : X \to Z : \Delta _0(x) = \Delta (\theta (x) \cdot x)$ . We then define the Zimmer $1$ -cocycle $\Omega : G \times X \to \Gamma $ such that $\Delta _0(g \cdot x) = \Omega (g,x) \cdot \Delta _0(x)$ for all $g \in G$ and a.e. $x \in X$ .

To translate the cocycle superrigidity Theorem 5.1 to an OE-superrigidity theorem, we use the connection with measure equivalence as developed in [Reference FurmanFur98, §3] (see also [Reference Drimbe and VaesDV21, Lemma 2.2] for a result that exactly suits our purposes). Define the action

(5.6) $$ \begin{align} G \curvearrowright X \times \Gamma : g \cdot (x,b) = (g \cdot x, \Omega(g,x) b), \end{align} $$

which commutes with the right translation action by $\Gamma $ in the second variable. By the results cited above, the action $G \curvearrowright X \times \Gamma $ admits a fundamental domain and there is a natural isomorphism of $\Gamma $ -actions $\alpha : G \backslash (X \times \Gamma ) \to Z$ with the property that $\Delta (x) \in \Gamma \cdot \alpha (x,e)$ for a.e. $x \in \mathcal {U}$ .

By Theorem 5.1, we find a group homomorphism $\delta : G \to \Gamma $ and a $1$ -cocycle $\gamma : \mathbb {R} \times Y \to C_\Gamma (\delta (G))$ such that $\Omega $ is cohomologous with the $1$ -cocycle

$$ \begin{align*}\Omega_1 : G \times X \to \Gamma : \Omega_1(g,x) = \delta(g) \, \gamma(-\omega(g,x),\psi(x)).\end{align*} $$

Let $\varphi : X \to \Gamma $ be a Borel map such that $\Omega _1(g,x) = \varphi (g \cdot x) \, \Omega (g,x) \, \varphi (x)^{-1}$ . The map $(x,b) \mapsto (x,\varphi (x)b)$ implements an isomorphism between the action $G \curvearrowright X \times \Gamma $ in (5.6) and the action

(5.7) $$ \begin{align} G \curvearrowright X \times \Gamma : g \cdot (x,b) = (g \cdot x, \Omega_1(g,x) b). \end{align} $$

Moreover, the action and the isomorphism commute with the $\Gamma $ -action. We thus still find an isomorphism of $\Gamma $ -actions $\alpha _1 : G \backslash (X \times \Gamma ) \to Z$ with the property that $\Delta (x) \in \Gamma \cdot \alpha _1(x,e)$ for a.e. $x \in \mathcal {U}$ .

The $1$ -cocycle $\widetilde {\Omega }_1 : G \times \widetilde {X} : \widetilde {\Omega }_1(g,(x,s)) = \Omega _1(g,x)$ for the Maharam extension $G \curvearrowright \widetilde {X}$ is, by construction, cohomologous with the $1$ -cocycle $(g,(x,s)) \mapsto \delta (g)$ . Since the action $G \curvearrowright X \times \Gamma $ admits a fundamental domain, a fortiori, the same holds for the action $G \curvearrowright \widetilde {X} \times \Gamma : g \cdot (x,s,b) = (g \cdot (x,s), \delta (g)b)$ , and thus for the action $\operatorname {Ker} \delta \curvearrowright \widetilde {X}$ . Since we assumed that a.e. action $G \curvearrowright \widetilde {X}_y$ is simple, the normal subgroup $\operatorname {Ker} \delta $ must be trivial. So $\delta : G \to \Gamma $ is faithful. Define $\Lambda = C_\Gamma (\delta (G))$ . Since G has trivial center, $\delta (G) \cap \Lambda = \{e\}$ . We have thus found a subgroup $\delta (G) \times \Lambda < \Gamma $ .

Since $\Omega _1$ takes values in $\delta (G) \times \Lambda $ , the action $\Gamma \curvearrowright G \backslash (X \times \Gamma )$ is induced from $\delta (G) \times \Lambda \curvearrowright G \backslash (X \times \delta (G) \times \Lambda )$ . Under the natural identification $G \backslash (X \times \delta (G) \times \Lambda ) = X \times \Lambda $ and the isomorphism $\delta \times \mathord {\mathrm {id}} : G \times \Lambda \to \delta (G) \times \Lambda $ , this last action is precisely the action given by (5.5).

Proof. Since $G < \operatorname {SL}(n,\mathbb {R})$ is dense, it follows from Lemma 4.5 that the action $G \curvearrowright \mathbb {R}^n$ is ergodic and not induced. By density of $G < \operatorname {SL}(n,\mathbb {R})$ , the center of G belongs to the center of $\operatorname {SL}(n,\mathbb {R})$ , which is trivial if n is odd. If n is odd and $\Sigma \lhd G$ is a normal subgroup whose action on $\mathbb {R}^n$ admits a fundamental domain, then the closure $\overline {\Sigma }$ of $\Sigma $ in $\operatorname {SL}(n,\mathbb {R})$ is, by density of G, a normal subgroup of $\operatorname {SL}(n,\mathbb {R})$ . Since n is odd, it follows that either $\overline {\Sigma } = \{1\}$ or $\overline {\Sigma } = \operatorname {SL}(n,\mathbb {R})$ . In the second case, $\Sigma $ is a dense subgroup of $\operatorname {SL}(n,\mathbb {R})$ , so that $\Sigma \curvearrowright \mathbb {R}^n$ is ergodic. It thus follows that $\Sigma = \{1\}$ .

By Theorem 4.4, $G < \operatorname {SL}(n,\mathbb {R})$ is essentially cocycle superrigid with countable targets. By [Reference Drimbe and VaesDV21, Proposition 3.3], the action $G \curvearrowright \mathbb {R}^n$ is cocycle superrigid with countable targets.

By [Reference Hahn and O’MearaHOM89, Proposition 4.3.11], the group $E(n,\mathcal {A})$ is finitely generated when $n \geq 3$ and $\mathcal {A}$ is finitely generated as a ring.