Proof. As observed in § 2.1, up to replacing $c$ by a cohomologous cocycle, and $X$ by a conull $G$-invariant Borel subset, we can (and do) assume that $c(A\times X)\subseteq B$. Likewise, up to replacing $Y$ by a conull $H$-invariant subset, we can choose an SOE cocycle $c':H\times Y\to G$ associated to the stable orbit equivalence $f^{-1}:V\to U$ between $H\curvearrowright Y$ and $G\curvearrowright X$, so that $c'(B\times Y)\subseteq A$.

Let $\Sigma =X\times H$, equipped with the measure-preserving action of $G\times H$ given by $(g,h)\cdot (x,k)= (gx,c(g,x)kh^{-1})$. Letting $X_e=X\times \{e\}$ (which is a fundamental domain for the $H$-action on $\Sigma$), we observe that $A X_e\subseteq B X_e$, so $A B X_e=BAX_e\subseteq B BX_e= BX_e$, thus $BX_e$ is invariant under $A\times B$. In addition, from the ergodicity of the $A$-action on $X$, we deduce that the action of $A\times B$ on $BX_e$ is ergodic. Moreover, for every $h\in H$, we have $hBX_e=BX_e$ if and only if $h\in B$, and otherwise $hBX_e\cap BX_e=\emptyset$. In addition, the union of all $H$-translates of $BX_e$ cover $\Sigma$. This proves that $\bar {H}=H/B$ acts simply transitively on the space $\bar \Sigma$ of ergodic components of the action of $A\times B$ on $\Sigma$.

By [Reference FurmanFur99b, Theorem 3.3], the space $\Sigma$ is measurably isomorphic to $Y\times G$, equipped with the measure-preserving action of $G\times H$ given by $(g,h)\cdot (y,k)=(hy,c'(h,y)kg^{-1})$. A symmetric argument then shows that $\bar {G}=G/A$ also acts simply transitively on $\bar {\Sigma }$.

Therefore, there exist an isomorphism $\alpha :\bar {G}\to \bar {H}$, and a measurable isomorphism $\bar {\Sigma }\approx \bar {H}$ sending $BX_e$ to $e$, such that the action of $\bar {G}\times \bar {H}$ on $\bar {\Sigma }$ is given by $(\bar {g},\bar {h})\cdot \bar {k}=\alpha (\bar {g})\bar {k}\bar {h}^{-1}$ through this identification. We also have a $(G\times H)$-equivariant Borel map $\Phi :\Sigma \to \bar {H}$ (sending $BX_e$ to $e$).

The equivariance of $\Phi$ shows that for all $g\in G$ and almost every $x\in X$, one has $\Phi (g(x,e))=\alpha (\bar {g})$, i.e. $\Phi (gx,c(g,x))=\alpha (\bar {g})$. Letting $h\in H$ be such that $\alpha (\bar {g})=\bar {h}$, we deduce that $\Phi (gx,c(g,x)h^{-1})=e$. This shows that $c(g,x)h^{-1}\in B$, i.e. $c(g,x)\in \alpha (gA)$, as desired.

Proof. Consider a partition $X=\sqcup _{i\in I}X_i$ into at most countably many Borel subsets such that for every $i\in I$, there exists a parabolic subgroup $P_i$ such that $(\mathcal {A}_{|X_i},\rho )$ is tightly $P_i$-supported (given by Lemma 5.1). As $\mathcal {A}$ is stably normalized by $\mathcal {H}$, Lemma 5.2 ensures that up to replacing $X$ by a conull Borel subset and refining the above partition, we can assume that $\mathcal {H}_{|X_i}\subseteq \rho ^{-1}(P_i\times P_i^{\perp })_{|X_i}$ for every $i\in I$. As $\mathcal {A}$ is stably normalized by $\mathcal {H}$ which is everywhere nonamenable, and as $\rho$ has trivial kernel, it follows from [Reference Horbez and HuangHH22, Lemma 3.10] that $P_i^\perp$ is nonamenable, and $(\mathcal {H}\cap \rho ^{-1}(P_i))_{|X_i}$ is amenable.

Proof. In this proof, we fix an identification between $G$ and $G_\Gamma$; parabolic subgroups of $G$ are understood with respect to this identification.

We first prove that assertion (i) implies property (ii)(a). For every $i\in I$, the group $P_i$ is nonabelian (Lemma 1.3); hence, $P_i$ splits as a direct product $P_i=M_i\times N_i$, where $M_i$ and $N_i$ are infinite parabolic subgroups, and at least one of them (say $N_i$) is nonabelian. Therefore, $N_i$ contains a nonabelian free subgroup. Choose an infinite cyclic subgroup $A_i\subseteq M_i$. Then $A_i$ commutes with $N_i$. The conclusion follows by letting $\mathcal {A}$ be a measured subgroupoid of $\mathcal {G}$ such that for every $i\in I$, one has $\mathcal {A}_{|X_i}=\rho ^{-1}(A_i)_{|X_i}$, and letting $\mathcal {N}$ be such that for every $i\in I$, one has $\mathcal {N}_{|X_i}=\rho ^{-1}(N_i)_{|X_i}$.

We now claim that if a measured subgroupoid $\mathcal {H}\subseteq \mathcal {G}$ satisfies property (ii)(a), then there exist a conull Borel subset $X^*\subseteq X$, a Borel partition $X^*=\sqcup _{i\in I}X_i$ into at most countably many Borel subsets, and for every $i\in I$, a join parabolic subgroup $P_i\subseteq G$ such that $\mathcal {H}_{|X_i}\subseteq \rho ^{-1}(P_i)_{|X_i}$. Once we prove the claim, we explain in the last paragraphs why this suffices to establish the lemma.

By Lemma 5.3, there exists a conull Borel subset $X^*\subseteq X$ and a partition $X^*=\sqcup _{i\in I}X_i$ into at most countably many Borel subsets such that for every $i\in I$, there exists a parabolic subgroup $R_i$ with $R_i^{\perp }$ nonabelian, such that $\rho (\mathcal {A}_{|X_i})\subseteq R_i$ and $\rho (\mathcal {N}_{|X_i})\subseteq R_i\times R_i^\perp$. Note that $R_i$ is nontrivial because $\mathcal {A}$ is of infinite type and $\rho$ has trivial kernel. In particular, $R_i\times R_i^{\perp }$ is a join parabolic subgroup.

Up to a further partition, for every $i\in I$, there exists a nontrivial parabolic subgroup $S_i\subseteq R_i\times R_i^{\perp }$ such that $(\mathcal {N}_{|X_i},\rho )$ is tightly $S_i$-supported (Lemma 5.1). As $\mathcal {N}$ is stably normalized by $\mathcal {H}$, up to a further partition and restriction to a further conull Borel subset of $X$, we can assume that $\rho (\mathcal {H}_{|X_i})\subseteq S_i\times S_i^{\perp }$ by Lemma 5.2. Lemma 1.2 ensures that $S_i\times S_i^{\perp }$ is a join parabolic subgroup, which proves our claim.

We have already proved that assertion (i) implies property (ii)(a). To see that assertion (i) implies property (ii)(b), let $\mathcal {H}$ be a measured subgroupoid as in assertion (i) (coming with a partition $X^*=\sqcup _{i\in I}X_i$ and maximal join parabolic subgroups $P_i$), and let $\mathcal {H}'$ be as in property (ii)(b). The above claim ensures that up to passing to a further conull Borel subset and refining the above partition, we can assume that for every $i\in I$, there exists a join parabolic subgroup $Q_i$ such that $\mathcal {H}'_{|X_i}\subseteq \rho ^{-1}(Q_i)_{|X_i}$. As $\mathcal {H}$ is stably contained in $\mathcal {H}'$ and $\rho$ is action-type, we deduce that every element of $P_i$ has a power contained in $Q_i$, and therefore $P_i\subseteq Q_i$ by [Reference MinasyanMin12, Lemma 6.4]. By maximality of $P_i$, we have $P_i=Q_i$, from which it follows that $\mathcal {H}'$ is stably contained in $\mathcal {H}$, proving property (ii)(b).

We finally prove that assertion (ii) implies assertion (i), so let $\mathcal {H}$ be as in assertion (ii). The above claim shows that there exists a Borel partition $X^*=\sqcup _{i\in I}X_i$ of a conull Borel subset into at most countably many subsets such that for every $i\in I$, $\rho (\mathcal {H}_{|X_i})$ is contained in a join parabolic subgroup $P_i$. The maximality assumption (ii)(b) together with the implication $\rm (i)\Rightarrow (ii)(a)$ implies that $P_i$ is maximal whenever $X_i$ has positive measure. Indeed, if $P_i$ is contained in a join parabolic subgroup $P'_i$, then $\rho ^{-1}(P_i)_{|X_i}\subseteq \rho ^{-1}(P'_i)_{|X_i}$, so these two subgroupoids would have to be stably equal. As $\rho$ is action-type, this implies that every element of $P'_i$ has a power contained in $P_i$. By [Reference MinasyanMin12, Lemma 6.4], it follows that $P_i=P'_i$, which proves the maximality of $P_i$. Using again the maximality assumption, after passing to a conull subset and a countable partition, we have $\mathcal {H}_{|X_i}=\rho ^{-1}(P_i)_{|X_i}$.

Proof. The lemma is clear when $G$ is abelian, so we assume otherwise. In particular, $\mathcal {G}$ is everywhere nonamenable. As usual, we fix an identification between $G$ and $G_\Gamma$; parabolic subgroups are understood with respect to this identification.

We first observe that statement (i) implies property (ii)(a). Indeed, if $\mathcal {H}$ is a subgroupoid as in statement (i), then amenability of $Z$ ensures that $\mathcal {H}$ is amenable (using that $\rho$ has trivial kernel), and the fact that $Z$ is normal in $G$ ensures that $\mathcal {H}$ is stably normalized by $\mathcal {G}$.

We now claim that if $\mathcal {H}$ satisfies property (ii)(a), then there exists a partition $X^*=\sqcup _{i\in I}X_i$ of a conull Borel subset $X^*\subseteq X$ into at most countably many Borel subsets such that for every $i\in I$, one has $\mathcal {H}_{|X_i}\subseteq \rho ^{-1}(Z)_{|X_i}$. Together with the maximality assertion (ii)(b) and the fact that a subgroupoid as in statement (i) satisfies property (ii)(a), this will show that $\rm (ii)\Rightarrow (i)$. This claim will also prove that every subgroupoid as in statement (i) is stably maximal with respect to property (ii)(a), showing that $\rm (i)\Rightarrow (ii)$.

We are thus left with proving the above claim. By Lemma 5.3, there exist a conull Borel subset $X^*\subseteq X$, a partition $X^*=\sqcup _{i\in I}X_i$ into at most countably many Borel subsets, and for every $i\in I$, a parabolic subgroup $P_i\subseteq G$ (with respect to the chosen standard generating set), such that:

1. (i) $\mathcal {H}_{|X_i}\subseteq \rho ^{-1}(P_i)_{|X_i}$,

2. (ii) $\mathcal {G}_{|X_i}\subseteq \rho ^{-1}(P_i\times P_i^{\perp })_{|X_i}$,

3. (iii) $\rho ^{-1}(P_i)_{|X_i}$ is amenable.

As $\rho$ is action-type, the second point implies that every element of $G$ has a power contained in $P_i\times P_i^{\perp }$, which, in turn, implies that $G=P_i\times P_i^{\perp }$ by [Reference MinasyanMin12, Lemma 6.4]. As $\rho$ is action-type and $\rho ^{-1}(P_i)_{|X_i}$ is amenable, the parabolic subgroup $P_i$ does not contain any nonabelian free subgroup, so it is abelian. These two facts together imply that $P_i\subseteq Z$, and the first point above completes our proof.

Proof. We prove the contrapositive statement, so assume that the center $Z_2$ of $G_2$ is nontrivial. Then $\mathcal {Z}=\rho _2^{-1}(Z_2)$ is a subgroupoid of $\mathcal {G}$ of infinite type which satisfies assertion 2 from Lemma 5.5 (by using the implication $\rm (i)\Rightarrow (ii)$ of that lemma, applied to the cocycle $\rho _2$). Using now the implication $\rm (ii)\Rightarrow (i)$ from Lemma 5.5, applied to the cocycle $\rho _1$, we deduce that there exists a Borel subset $U\subseteq Y$ of positive measure such that $\rho _1(\mathcal {Z}_{|U})$ is contained in the center $Z_1$ of $G_1$. As $\rho _1$ has trivial kernel and $\mathcal {Z}$ is of infinite type, this implies that $Z_1$ is nontrivial.

Proof. Let $i\in I$ be such that $X_i$ has positive measure, and let $C_i,C'_i$ be the respective centers of $P_i,P'_i$. Let $\hat {\mathcal {Z}}_i=\rho ^{-1}(C_i)_{|X_i}$ and $\hat {\mathcal {Z}}'_i=\rho ^{-1}(C'_i)_{|X_i}$. Note that $\mathcal {H}_{|X_i}$ and $\mathcal {H}'_{|X_i}$ admit strict action-type cocycles towards $P_i,$ and $P'_i$, respectively. Therefore, Lemma 5.5 ensures that $\hat {\mathcal {Z}}_i$ and $\hat {\mathcal {Z}}'_i$ are central subgroupoids of $\mathcal {H}_{|X_i}$ and $\mathcal {H}'_{|X_i}$, respectively, and, conversely, every central subgroupoid of $\mathcal {H}_{|X_i}$ or $\mathcal {H}'_{|X_i}$ is stably equal to $\hat {\mathcal {Z}}_i$ or $\hat {\mathcal {Z}}'_i$, respectively.

Assuming that assertion (i) holds, the group $\langle C_i,C'_i\rangle$ is abelian. Let $\mathcal {Z}_i$ and $\mathcal {Z}'_i$ be central subgroupoids of $\mathcal {H}_{|X_i}$ and $\mathcal {H}'_{|X_i}$, respectively. Let $U\subseteq X_i$ be a Borel subset of positive measure, and let $V\subseteq U$ be a Borel subset of positive measure such that $(\mathcal {Z}_i)_{|V}=(\hat {\mathcal {Z}}_i)_{|V}$ and $(\mathcal {Z}'_i)_{|V}=(\hat {\mathcal {Z}}'_i)_{|V}$. Then $\langle (\mathcal {Z}_i)_{|V},(\mathcal {Z}'_i)_{|V}\rangle \subseteq \rho ^{-1}(\langle C_i,C'_i\rangle )_{|V}$ is amenable (as $\rho$ has trivial kernel). It follows that assertion (ii) holds.

Assuming that assertion (i) fails, there exist infinite cyclic subgroups $A_i\subseteq C_i$ and $A'_i\subseteq C'_i$ that together generate a rank-two free group: this follows, for instance, from [Reference Kim and KoberdaKK14, Theorem 44]. Let $V\subseteq X_i$ be any Borel subset of positive measure. It follows from [Reference KidaKid10, Lemma 3.20] that $\langle \rho ^{-1}(A_i)_{|V},\rho ^{-1}(A'_i)_{|V}\rangle$ is nonamenable. Therefore, $\langle (\hat {\mathcal {Z}}_i)_{|V},(\hat {\mathcal {Z}}'_i)_{|V}\rangle$ is nonamenable, showing that assertion (ii) fails.

Proof. Throughout the proof, we always identify $G,H$ with right-angled Artin groups $G_\Gamma,G_\Lambda$, in such a way that through these identifications, all standard generators act ergodically on $X,Y$. All parabolic subgroups will be understood with respect to these identifications.

The groupoid $\mathcal {G}=(G\ltimes X)_{|U}$ is naturally isomorphic (via $f$) to $(H\ltimes Y)_{|V}$ (after renormalizing the measures on $U$ and $V$). Thus, $\mathcal {G}$ comes equipped with two action-type cocycles $\rho _G:\mathcal {G}\to G$ and $\rho _H:\mathcal {G}\to H$.

Let $P\subseteq G$ be a maximal join parabolic subgroup with trivial center, and let $\mathcal {P}=\rho _G^{-1}(P)$. Then $\mathcal {P}$ satisfies assertion (ii) from Lemma 5.4, as it follows from applying this lemma to the cocycle $\rho _G$. Using now the implication $\rm (ii)\Rightarrow (i)$ from Lemma 5.4, applied to the cocycle $\rho _H$, we see that $\mathcal {P}$ also satisfies assertion (i) with respect to $\rho _H$. In particular, there exist a Borel subset $W\subseteq U$ of positive measure and a maximal join parabolic subgroup $Q\subseteq H$ such that $\mathcal {P}_{|W}=\rho _H^{-1}(Q)_{|f(W)}$. In particular, $\mathcal {P}_{|W}$ comes equipped with two action-type cocycles towards $P$ and $Q$. As $P$ has trivial center, Corollary 5.6 ensures that $Q$ also has trivial center.

Let $c:G\times X\to H$ be an SOE cocycle associated to $f_{|W}$. Note that $c$ is also an SOE cocycle associated to $f$, and any two such cocycles are cohomologous. Thus, if we prove that $c$ is cohomologous to a group isomorphism, then the same is true of any SOE cocycle associated to $f$. Note also that $\kappa (f_{|W})=\kappa (f)$.

The actions $P\curvearrowright X$ and $Q\curvearrowright Y$ are ergodic (by our irreducibility assumption), and the above ensures that for almost every $x\in W$, one has $f((P\cdot x)\cap W)=(Q\cdot f(x))\cap f(W)$. Thus, $c$ is cohomologous to a cocycle $c'$ such that $c'_{|P\times X}$ is an SOE cocycle associated to $f_{|W}$ for the stable orbit equivalence between $P\curvearrowright X$ and $Q\curvearrowright Y$ (in particular $c'(P\times X^*)\subseteq Q$ for some conull Borel subset $X^*\subseteq X$). The groups $P$ and $Q$ are join parabolic subgroups with trivial center, so they split as direct products $P=P_{1}\times \dots \times P_{k}$ and $Q=Q_1\times \dots \times Q_\ell$ of at least two nonabelian parabolic subgroups that do not admit any nontrivial product decomposition. All subgroups $P_i$ and $Q_j$ belong to Monod and Shalom's class $\mathcal {C}_{\mathrm {reg}}$, so Theorem 3.2 ensures that $\kappa (f_{|W})=1$ (so $\kappa (f)=1$) and that $c'_{|P\times X}$ is cohomologous to a group isomorphism $\alpha :P\to Q$. As $P$ contains a conjugate of a standard generator of $G$, Lemma 4.3 ensures that $c$ is cohomologous to a group isomorphism $\alpha :G\to H$, and the actions $G\curvearrowright X$ and $H\curvearrowright Y$ are conjugate through $\alpha$. As already mentioned in the previous paragraph, this is enough to conclude.

Proof. As in our previous proof, we always identify $G,H$ with right-angled Artin groups $G_\Gamma,G_\Lambda$, in such a way that through these identifications, all standard generators act ergodically on $X,Y$. All parabolic subgroups will be understood with respect to these identifications.

Up to exchanging the roles of $G$ and $H$, we assume without loss of generality that $\kappa (f)\ge 1$. Let $\mathcal {G}=(G\ltimes X)_{|U}$, which is naturally isomorphic (through $f$) to $(H\ltimes Y)_{|V}$ (after renormalizing the measures on $U$ and $V$). Then $\mathcal {G}$ comes equipped with two action-type cocycles $\rho _G:\mathcal {G}\to G$ and $\rho _H:\mathcal {G}\to H$.

Lemma 1.4 ensures that there exist two distinct maximal join parabolic subgroups $P_1,P_2\subseteq G$, whose centers $A_1,A_2$ are infinite and commute, and by Lemma 1.5 we have $A_1\cap A_2=\{1\}$.

For every $i\in \{1,2\}$, let $\mathcal {P}_i=\rho _G^{-1}(P_i)$ and $\mathcal {A}_i=\rho _G^{-1}(A_i)$. Then $\mathcal {P}_1$ and $\mathcal {P}_2$ are two maximal join subgroupoids of $\mathcal {G}$ which contain a central subgroupoid of infinite type, and are center-commuting. Using Lemmas 5.4 and 5.7, we can therefore find two distinct maximal join parabolic subgroups $Q_1,Q_2\subseteq H$ with commuting infinite centers $B_1,B_2$, and a Borel subset $W\subseteq U$ of positive measure, such that for every $i\in \{1,2\}$, one has $(\mathcal {P}_i)_{|W}=\rho _H^{-1}(Q_i)_{|f(W)}$. In addition, Lemma 5.5 implies that up to replacing $W$ by a positive measure Borel subset, we can assume that $(\mathcal {A}_i)_{|W}=\rho _H^{-1}(B_i)_{|f(W)}$ for every $i\in \{1,2\}$.

Let $c:G\times X\to H$ be an SOE cocycle associated to $f_{|W}$. Up to replacing $X$ by a conull invariant Borel subset, we can (and do) assume that $c$ is chosen so that whenever $(g,x)\in G\times X$ satisfies $x,gx\in W$, then $c(g,x)$ is the unique element $h\in H$ so that $f(gx)=hf(x)$. We prove that $c$ is cohomologous to a cocycle $c'$ for which there exists a standard generator $s\in G$ such that $c'(s,\cdot )$ is almost everywhere constant. This will be enough to conclude the proof of our proposition in view of Lemma 4.3 (after observing that $c$ is also an SOE cocycle associated to $f$, and $\kappa (f_{|W})=\kappa (f)$).

For $i\in \{1,2\}$, we have $(\mathcal {P}_i)_{|W}=\rho _H^{-1}(Q_i)_{|f(W)}$, and $P_i,Q_i$ act ergodically on $X,Y$ by assumption. Thus, $c$ is cohomologous to a cocycle $c_i$ such that $(c_i)_{|P_i\times X}$ is an SOE cocycle associated to $f_{|W}$ for the stable orbit equivalence between $P_i\curvearrowright X$ and $Q_i\curvearrowright Y$, and such that $c$ and $c_i$ coincide on all pairs $(g,x)$ with $x,gx\in W$.

By Lemma 3.1 (applied to the ambient groups $P_i,Q_i$ and to the normal subgroups $A_i,B_i$), for every $i\in \{1,2\}$, up to replacing $c_i$ by a cohomologous cocycle and replacing $X$ by a conull $G$-invariant and $H$-invariant Borel subset, we can assume that the following hold:

1. (i) there is a group isomorphism $\bar {\alpha }_i:P_i/A_i\to Q_i/B_i$ satisfying that for every $g\in P_i$ and every $x\in X$, one has $c_i(g,x)\in \bar {\alpha }_i(gA_i)$ and in particular, $c_i(A_i\times X)\subseteq B_i$;

2. (ii) $c_i$ coincides with $c$ on all pairs $(g,x)$ with $x,gx\in W$.

Let $r_i:Q_i=B_i\times B_i^{\perp }\to B_i^{\perp }$ be the retraction, and let $c'_i:A_i^{\perp }\times X\to B_i^{\perp }$ be the cocycle defined as $c'_i=(r_i\circ c_i)_{|A_i^{\perp }\times X}$. There are isomorphisms $P_i/A_i\to A_i^{\perp }$ and $Q_i/B_i\to B_i^{\perp }$ (coming from choosing the unique lift). Through these identifications $\bar {\alpha }_i$ yields an isomorphism $\alpha _i:A_i^{\perp }\to B_i^{\perp }$ such that for every $g\in A_i^{\perp }$ and every $x\in X$, one has $c'_i(g,x)=\alpha _i(g)$.

Recall from Lemma 1.5 that for every $i\in \{1,2\}$, we have $A_{3-i}\subseteq A_i^{\perp }$ and $B_{3-i}\subseteq B_i^{\perp }$. We now prove that for every $i\in \{1,2\}$, the isomorphism $\alpha _i$ restricts to an isomorphism between $A_{3-i}$ and $B_{3-i}$. By symmetry, it suffices prove it for $i=2$. By Poincaré recurrence, for every $g\in A_1$, there exist an integer $n>0$ and $x\in W$ such that $g^nx\in W$. Then $c_2(g^n,x)=c_1(g^n,x)$ (they are both equal to $c(g^n,x)$), and these belong to $B_1$, which is contained in $B_2^{\perp }$. In particular, $c_2(g^n,x)=c'_2(g^n,x)$, which, in turn, equals $\alpha _2(g)^n$ by the above. Thus, $\alpha _2(g)^n\in B_1$, and therefore $\alpha _2(g)\in B_1$ by [Reference MinasyanMin12, Lemma 6.4]. Thus, $\alpha _2(A_1)\subseteq B_1$. We now show that actually $\alpha _2(A_1)=B_1$. Take $h\in B_1$. Then there exist an integer $m>0$ and $y\in f(W)$ such that $h^my\in f(W)$. As the actions $A_1\curvearrowright X$ and $B_1\curvearrowright Y$ induce (via $f$) the same orbit equivalence relation on $W$, there exists $g\in A_1$ such that $c_1(g,y)=c_2(g,y)=h^m$. As $h^m\in B_1\subseteq B_2^{\perp }$, we have $c_2(g,y)=c'_2(g,y)$, so $\alpha _2(g)=h^m$. As $\alpha _2:A_2^\perp \to B_2^\perp$ is an isomorphism, there also exists $g_0\in A_2^\perp$ such that $\alpha _2(g_0)=h$, hence $g^m_0=g\in A_1$. It follows that $g_0\in A_1$ by [Reference MinasyanMin12, Lemma 6.4]. This proves that $\alpha _2$ restricts to an isomorphism between $A_1$ and $B_1$, as desired.

For every $i\in \{1,2\}$, we can therefore extend $\alpha _i$ on $A_i$ by defining $(\alpha _i)_{|A_i}=(\alpha _{3-i})_{|A_i}$ (in particular, $\alpha _1$ and $\alpha _2$ coincide on $\langle A_1,A_2\rangle$). This yields an isomorphism $\alpha _i:P_i\to Q_i$ such that for every $g\in A_i\cup A_i^{\perp }$ and every $x\in X$, one has $c_i(g,x)\in \alpha _i(g)B_i$. Now, using the cocycle relation and the fact that every element of $P_i$ is a product of the form $hk$ with $h\in A_i$ and $k\in A_i^{\perp }$, we see that $c_i(g,x)\in \alpha _i(g)B_i$ for every $g\in P_i$ and almost every $x\in X$.

Altogether, these show that, for every $i\in \{1,2\}$, we have a measurable map $\varphi _i:X\to H$ and a measurable map $\kappa _i:P_i\times X\to B_i$ such that for every $g\in P_i$ and almost every $x\in X$, one has $c(g,x)=\varphi _i(gx)\alpha _i(g)\kappa _i(g,x)\varphi _i(x)^{-1}$. Up to replacing $X$ by a conull $G$-invariant Borel subset, we assume that these relations hold for every $g\in G$ and every $x\in X$. Let $X=\sqcup _{j\in J}X_j$ be a partition into at most countably many Borel subsets such that for every $j\in J$, the maps $\varphi _1,\varphi _2$ have constant values $\gamma _{1,j},\gamma _{2,j}$ when restricted to $X_j$.