# Groupoid exercises and modular consequences A change of section changes a return cocycle by a coboundary. A change of measure does the same to a modular cocycle. These two calculations connect suspension flows, groupoid structure and the domains of modular operators. This lesson solves the whole exercise set following Takesaki III, XIII §3, including the cases where an assertion needs a correction. Arrows are written \((y,x):x\to y\), with the rightmost arrow traversed first. Measured statements use a common invariant conull Borel reduction. A finite relation has finite classes, with no uniform bound on their sizes. “Hyperfinite” means an increasing exhaustion by such full-unit relations. A labelled transformation groupoid retains stabilizers; its principal orbit relation forgets them. The precise earlier programme inputs are the finite cochain correction, 2. Small corrections that preserve old errors, stabilizer splitting, 3. Splitting the full countable groupoid, relation weight model, 1. Counting arrows and fixing the derivative through 2. The weight model at infinite mass, and the following proved constructions: dense roofs and eigenfunctions, the full two-gap construction, TG §§1–12, irrational rotation stages §§1–4, cocycle groupoids §§1–5, and localization, including uniform cohomology, §§1–7. We give the applications and corrections below with their full calculations. ## 1. Dense groups and positive roofs **Small cochain theorem.** Let \(R\) be a hyperfinite principal relation and \(p:R\to\mathbb R\) a Borel additive cocycle. For every dense subgroup \(H\leq\mathbb R\) and every \(\varepsilon>0\), there is a Borel \(c:X\to\mathbb R\), with \(|c|<\varepsilon\), such that \[ p^c(y,x)=p(y,x)+c(y)-c(x)\in H. \tag{GEX.1} \] No Borel hypothesis on \(H\) is needed. Choose one member of \(H\) in each rational interval that meets it; the subgroup \(H_0\) generated by this countable set is countable and dense. Apply CLR-02 with \(P=\mathbb R\), subgroup \(H_0\), first cocycle zero and second cocycle \(p\). On the first finite relation obtain \[ p=h_1+f_1(y)-f_1(x),\qquad |f_1|<\varepsilon/4. \] Extend to successive finite relations with \(|f_{n+1}-f_n|<\varepsilon2^{-n-2}\), preserving the old \(h_n\) exactly. The uniform Cauchy limit \(f\) is Borel and has absolute value less than \(\varepsilon/2\). On each arrow the error eventually stops changing, so it remains in \(H_0\), rather than merely in its closure. Passing to the limit gives (GEX.1) with \(c=-f\). For an ergodic nonsingular transformation \(S\), its principal relation is hyperfinite by the tower proof in the earlier roof lesson, Proposition 1.2. When \(S\) is aperiodic, \((n,x)\mapsto(S^nx,x)\) also identifies the labelled groupoid with that relation. Periodic labels require a separate answer. On a \(k\)-cycle choose its Borel least point \(x_0\) and write \(x_i=S^ix_0\). A labelled cocycle has period \(P=p(k,x_0)\). Every gauge leaves \(P\) unchanged. Conversely set \(c(x_i)=-p(i,x_0)\). The adjusted successive labels are zero except for the last label \(P\). Thus an \(H\)-valued representative exists exactly when \(P\in H\). In particular the one-point action with \(p(n,*)=n\sqrt2\) and \(H=\mathbb Q\) disproves an unrestricted assertion about periodic labelled groupoids. **Moving a section.** Suppose a nonsingular flow \(T_t\) is represented by a Borel suspension over \(S:Z\to Z\), with finite roof \(r\geq\delta>0\), and measure class \(\nu(dz)\,dv\) on \(0\leq v0), \] with negative sums for \(n<0\). Choose the preceding \(c\) with \(|c|<\delta/4\). Move the point \(z\) on its flow orbit to \(T_{c(z)}z\). Consecutive moved points have distance \[ r'(z)=r(z)+c(Sz)-c(z)>\delta/2,\qquad r'(z)\in H. \tag{GEX.2} \] The moved points remain ordered and bi-infinite: their displacement is bounded and the old gaps are at least \(\delta\). Consequently \[ (z,v)\longmapsto T_{c(z)+v}z,\qquad 0\leq v0\), \(a/b\notin\mathbb Q\), density follows directly: a nondense additive subgroup has a positive least positive element and is cyclic, which would make \(a/b\) rational. To justify the dichotomy, if the positive elements have infimum zero, integer multiples of an arbitrarily small one approximate any real number; otherwise the positive infimum is attained by taking differences of a minimizing sequence, and Euclidean division proves cyclicity. For an ergodic periodic flow of period \(P>0\), an \(H\)-valued positive roof exists precisely when \(P\in H\). The constant roof \(P\) over one point proves sufficiency. Necessity follows by summing the finitely many positive return intervals around the periodic orbit. There are finitely many because a suspension has divergent forward and backward return sums. The fixed one-point flow has no positive-roof realization: translation by a sufficiently small positive time moves a positive-measure subinterval of any positive-height tube. ## 2. Two ceilings and the constant-roof criterion **Two prescribed ceilings.** If \(a,b>0\) and \(a/b\notin\mathbb Q\), every free ergodic nonsingular real flow is a suspension with roof in \(\{a,b\}\). The complete earlier TG proof gives, for every free Borel real action, a Borel section on every orbit with consecutive gaps \(a\) or \(b\). Its §§2–9 construct separated sections and convergent finite block corrections; §§10–12 treat the remaining sparse and endpoint orbits and prove the exact half-open suspension bijection. Thus it supplies an actual proved programme input, including all orbits. It is stronger than asking for a dense-group-valued roof: a positive element of \(a\mathbb Z+b\mathbb Z\) need not be a nonnegative combination of \(a,b\). Here is the measure step, also proved in the roof lesson, Lemma 2.1. More generally let the section roof be finite and bounded below by \(\delta>0\). Replace the flow measure by an equivalent probability \(m\), and let \(\nu\) be its marginal under the Borel base projection. For a Borel base set \(A\), its full tube is null exactly when its lower tube \(A\times[0,\delta/2)\) is null: rational translates of the lower tube cover the full tube and the reverse inclusion is immediate. Flow nonsingularity then makes the marginal base transformation nonsingular. Indeed countably many rational translates of the \(A\)-tube cover the \(SA\)-tube: from any point of the latter, the times moving it into the former contain a nonempty open interval. Apply the same argument to \(S^{-1}\). Choose a strictly positive integrable function \(w\) on \(\mathbb R\), of integral one, and average \(m\) under time translations. The resulting measure \(\bar m\) is equivalent to \(m\), since every translated \(m\) is equivalent to \(m\). On a Borel set \(B\) the orbit-coordinate formula is \[ \bar m(B)=\int\sum_{k\in\mathbb Z} \int_0^{r(S^kz)} w\bigl(R_k(z)+v-u\bigr)\, 1_B(S^kz,v)\,dv\,dm(z,u). \tag{GEX.4} \] The half-open return intervals partition the real line, proving this formula by change of variable in the time integral. If \(B\) is null for \(\nu(dz)\,dv\), its vertical sections are null for \(\nu\)-almost every \(z\). Base nonsingularity makes every summand zero for \(m\)-almost every \((z,u)\). Conversely, if \(B\) has positive product measure, its sections have positive length on a positive-\(\nu\) base set. The \(k=0\) summand is strictly positive above that set because \(w>0\), hence has positive \(m\)-integral. This proves \(\bar m\sim\nu(dz)\,dv\). A base invariant set gives an invariant tube, so flow ergodicity gives base ergodicity. This completes the measured two-ceiling assertion without restricting the ambient measure to the null section. The ergodic periodic exceptions are exact: \[ P=ma+nb,\qquad m,n\in\mathbb Z_{\geq0},\quad m+n>0. \tag{GEX.5} \] A finite cycle with \(m\) roofs \(a\) and \(n\) roofs \(b\) proves sufficiency; summing the gaps proves necessity. If both values must occur on positive-measure base sets, require \(m,n\geq1\). For example \(P=\sqrt2-1\) belongs to \(\mathbb Z+\sqrt2\mathbb Z\), but is smaller than either permitted gap, so the dense-group conclusion cannot imply two ceilings for this periodic flow. These cases exhaust ergodic real flows. The roof lesson, Lemma 5.1, proves the needed measurable trichotomy using the injective orbit map into the Polish space of measurable real-time paths. Stabilizers are closed subgroups of \(\mathbb R\); their period is a Borel invariant. Ergodicity gives respectively a free action, one circle orbit of period \(P\) with Lebesgue measure class, or a fixed point. In the periodic case the compact-orbit code is an invariant standard Borel code and is constant almost everywhere, which is why a single circle, rather than a nontrivial field of circles, occurs. **Constant roofs.** With \(\alpha_tu=u\circ T_{-t}\), an ergodic flow has a constant positive roof if and only if \[ \alpha_tu=e^{is_0t}u,\qquad |u|=1,\qquad s_0\ne0. \tag{GEX.6} \] The nonzero condition is essential: \(u=1,s_0=0\) exists for every flow. In a constant-roof suspension of height \(r\), the function \(u(z,v)=e^{2\pi iv/r}\) has frequency \(-2\pi/r\), proving one implication. Conversely put \(\omega=|s_0|\), \(r=2\pi/\omega\), and take \(v=u\) if \(s_0<0\), \(v=u^*\) otherwise. Then \(v(T_tx)=e^{i\omega t}v(x)\) for each fixed \(t\) almost everywhere. The earlier localization theorem, Theorem 1.2, repairs this equivariant measurable map into the circle to a Borel representative on one invariant conull set with the identity for every \(t\) and every \(x\). Put \[ \tau(x)=\omega^{-1}\operatorname{Arg}v(x)\in[0,r),\qquad z(x)=T_{-\tau(x)}x,\qquad Z=\{v=1\}. \] The map \(Z\times[0,r)\to X\), \((z,t)\mapsto T_tz\), has inverse \((z(x),\tau(x))\). The return map is \(S=T_r|_Z\). These formulas prove Borel bijectivity and intertwining of time, including periodic orbits. Apply (GEX.4) to obtain the base measure class and ergodicity. For a circle of period \(P\), the possible constant roofs are exactly \(P/k\), \(k\geq1\), by the finite-cycle criterion. ## 3. The exact irrational Weyl factor On \(L^2(\mathbb R)\), let \(U(a)\xi(t)=\xi(t+a)\), \(V(b)\xi(t)=e^{ibt}\xi(t)\), and assume \(\theta=ab/(2\pi)\) is irrational. Set \(u=V(b)\), \(v=U(a)\), and \(\lambda=e^{2\pi i\theta}\). Thus \[ vuv^*=\lambda u,\qquad \mathcal R=\{u,v\}''. \] This is the type \(II_1\) factor specified in the exercise. The earlier rotation-stage proof, Lemma 1.1, uses its faithful normal trace \(\tau\) to show \[ \tau(u^mv^n)=1_{\{m=n=0\}}. \tag{GEX.7} \] For completeness, conjugation by \(v\) multiplies this trace by \(\lambda^m\), and when \(m=0\), conjugation by \(u\) multiplies it by \(\lambda^{-n}\). Irrationality forces vanishing in every nontrivial case. The monomials form an orthonormal basis of the tracial GNS space: their algebra generates \(\mathcal R\), and bounded strong approximation gives \(L^2\) density. Their diagonal moments give Haar spectral measure for \(u\). Take \(T(z)=\lambda^{-1}z\) on the Haar circle and \(\alpha(f)=f\circ T^{-1}\). The action is free since \(\lambda^n\ne1\) for \(n\ne0\). Fourier coefficients of an invariant \(L^2\) function vanish at every nonzero integer, so it is ergodic. In the regular crossed product \(N=L^\infty(\mathbb T)\rtimes_\alpha\mathbb Z\), the coordinate and implementing unitaries \(u_0,v_0\) have the same trace moments as (GEX.7). The unitary between the two GNS spaces sending \(u_0^mv_0^n\) to \(u^mv^n\) intertwines left multiplication, since on both bases it is \[ u:(m,n)\mapsto(m+1,n),\qquad v:(m,n)\mapsto\lambda^m(m,n+1). \] Consequently it conjugates the generated von Neumann algebras. Faithful normal tracial GNS representations give a normal isomorphism \(\Phi:N\to\mathcal R\), carrying the whole diagonal to \(f(u)\). This is the earlier rotation-stage Theorem 2.1; matching polynomial relations alone would not prove normality. Identify \(N\) with the principal relation algebra through \((n,x)\mapsto(T^nx,x)\). The tower exhaustion in the earlier rotation-stage Corollary 4.2 gives finite full-unit relations \(F_n\uparrow R_T\) on an invariant conull set. Let \(N_n\) be generated in the full relation representation by all diagonal multipliers and partial orbit operators with graphs in \(F_n\). Here is its precise type I form. On the part with \(q\)-point classes, order the class over its least-point selector \(b\) as \(x_0(b),\ldots,x_{q-1}(b)\), and put \(\nu_q=q\,m|_{B_q}\). Invariance gives equal measure to every sheet. The unitary coordinates \[ (W_n\xi)(b)_{i,y}=q^{-1/2}\xi(x_i(b),y), \qquad y\in[b]_{R_T}, \] give the Hilbert field \(\mathbb C^q\otimes\ell^2([b]_{R_T})\) over \((B_q,\nu_q)\). Sheet maps are \(e_{ij}\otimes1\); diagonal functions supply every scalar function of \(b\) and all diagonal matrix entries. Thus \[ N_n\cong\prod_{q\geq1}L^\infty(B_q,\nu_q)\,\bar\otimes\,M_q(\mathbb C). \tag{GEX.8} \] This is faithful because each multiplicity space is nonzero, and normal because matrix entries preserve increasing bounded limits. Conversely every generator acts as one of these measurable matrix fields, so (GEX.8) is equality, as in the earlier Proposition 3.1. Put \(D_n=\{x:(Tx,x)\in F_n\}\). Then \(v_0 1_{D_n}\in N_n\), and \(1_{D_n}\uparrow1\) strongly. Therefore \(v_01_{D_n}\to v_0\) strongly and \[ \|v_0-v_01_{D_n}\|_{2,\tau_N}^{\,2}=m(D_n^c)\longrightarrow0. \] Every diagonal function already belongs to every \(N_n\). These generators show \[ \mathcal R=\left(\bigcup_n\Phi(N_n)\right)''. \tag{GEX.9} \] The single commutant in the printed exercise is impossible. If instead \(\mathcal R=(\bigcup_nM_n)'\) with \(M_n\subset\mathcal R\), each \(M_n\) would lie in \(\mathcal R\cap\mathcal R'=\mathbb C1\). Its union would then have commutant \(B(L^2(\mathbb R))\), contradicting type \(II_1\). The double commutant in (GEX.9) is the corrected statement. In this probability-preserving model the derivative cocycle is one. RWT-02 therefore gives \(S\xi(y,x)=\overline{\xi(x,y)}\), \(J=S\), and \(\Delta=1\), all on the whole relation Hilbert space. Every complex power of \(\Delta\) also has full domain. The type I approximants in (GEX.8) may have diffuse centres; nothing in their construction says they are ordinary finite-dimensional algebras. ## 4. Cocycle arrows and their topology Let \(G\) be locally compact and \(M\) a factor with separable predual. An object is a continuous action \(\alpha:G\to\operatorname{Aut}(M)\). An arrow \((a,\alpha)\) consists of a continuous unitary cocycle \[ a_{gh}=a_g\alpha_g(a_h),\qquad a_e=1. \] Its source is \(\alpha\), its range is \(\alpha^a_g=\operatorname{Ad}(a_g)\alpha_g\), and \[ (b,\alpha^a)(a,\alpha)=(ba,\alpha),\qquad (a,\alpha)^{-1}=(a^*,\alpha^a). \tag{GEX.10} \] Indeed \[ b_{gh}a_{gh} =b_g a_g\alpha_g(b_h)a_g^*a_g\alpha_g(a_h) =(b_ga_g)\alpha_g(b_ha_h). \] Also \((\alpha^a)^b=\alpha^{ba}\), and \(a_{gh}^*=a_g^*\alpha^a_g(a_h^*)\). These identities verify the range, inverse and cocycle laws; associativity is pointwise associativity of unitary multiplication and the unit at \(\alpha\) is \((1,\alpha)\). Use the strong-star topology on the unitary group and the usual standard-form topology on \(\operatorname{Aut}(M)\). The earlier cocycle-groupoid Lemma 1.1 proves both groups Polish and the maps \((\theta,u)\mapsto\theta(u)\), \(u\mapsto\operatorname{Ad}u\) continuous. Compact-open convergence and finite covers of compact images then make all operations in (GEX.10) continuous for arbitrary locally compact \(G\). For example, uniform convergence on a fixed compact subset of \(G\) lets continuity of multiplication be tested on finitely many neighbourhoods covering the limiting compact image. The same argument treats composition of an automorphism with a unitary-valued map. If \(G\) is **second countable**, the arrow space is Polish. Indeed \(C(G,Y)\) is Polish for Polish \(Y\), as proved in the earlier Lemma 1.2: choose a cofinal compact exhaustion \(K_n\) and a bounded complete metric \(d\) on \(Y\); the complete metric is \[ D(f,h)=\sum_{n\geq1}2^{-n}\sup_{g\in K_n}d(f(g),h(g)). \] A countable base is obtained from compact closures of a countable relatively compact base in \(G\) and a countable base in \(Y\), using finite compact-open intersections. Completeness follows by uniform limits on each \(K_n\), agreeing on overlaps. In \(C(G,\mathcal U(M))\times C(G,\operatorname{Aut}(M))\), the action identities, normalization and cocycle identities are closed equations for every \(g,h\). Their common solution set is closed, hence Polish. The objects form the analogous closed action space. This proves all the groupoid and topological assertions at the hypotheses needed for Polishness. Ordinary separability of \(G\) is insufficient. Let \(G=\{-1,1\}^{\mathbb R}\), compact in the product topology. It is separable: the functions taking value \(+1\) on a finite union of bounded rational intervals and \(-1\) elsewhere form a countable dense set, since they realize every prescribed finite pattern. For the trivial action on \(M_2(\mathbb C)\), each coordinate \(t\in\mathbb R\) gives a cocycle \(a^t(g)=g(t)1\). Distinct coordinates have uniform distance two on the compact domain \(G\). The open uniform balls of radius \(1/2\) around them are pairwise disjoint. Thus the compact-open cocycle space contains an uncountable disjoint family of open sets and is not separable, hence not Polish. Compactness here follows from the product compactness proof in the earlier Example 2.3. The groupoid formulas remain valid; it is the unrestricted Polish claim that fails. ## 5. Semidirect products and the correct source Let \(\mathcal G\) act on a groupoid \(\mathcal H\). Write \(\pi:\mathcal H\to\mathcal G^{(0)}\) for the fibre label, so each fibre is a subgroupoid, and let \(T_\gamma:\mathcal H_{s\gamma}\to\mathcal H_{r\gamma}\) be the isomorphism attached to \(\gamma\), with \(T_{\gamma_1\gamma_2}=T_{\gamma_1}T_{\gamma_2}\). Use source coordinates \[ \mathcal K=\{(\gamma,h):s\gamma=\pi(h)\},\qquad s(\gamma,h)=s_{\mathcal H}h,\quad r(\gamma,h)=T_\gamma(r_{\mathcal H}h). \tag{GEX.11} \] The objects are those of \(\mathcal H\), and the structure maps are \[ \begin{aligned} (\gamma_1,h_1)(\gamma_2,h_2) &=(\gamma_1\gamma_2,T_{\gamma_2}^{-1}(h_1)h_2),\\ (\gamma,h)^{-1}&=(\gamma^{-1},T_\gamma(h^{-1})),\\ 1_u&=(1_{\pi(u)},u). \end{aligned} \tag{GEX.12} \] The product is defined exactly when \(s_{\mathcal H}h_1=T_{\gamma_2}r_{\mathcal H}h_2\). Applying \(\pi\) makes \(\gamma_1,\gamma_2\) composable, and applying \(T_{\gamma_2}^{-1}\) makes the second-coordinate product composable. Its source and range are those of the proposed composite. Both ways of multiplying three arrows have second coordinate \[ T_{\gamma_2\gamma_3}^{-1}(h_1)\, T_{\gamma_3}^{-1}(h_2)\,h_3, \] so associativity holds. Multiplying either side by the displayed inverse gives the appropriate unit; the unit equations follow from \(T_{1}=1\). These checks prove the full groupoid law. For Borel structure maps and a Borel action, the fibre product is Borel and every displayed operation is Borel. For Polish groupoids, continuous \(\pi\) and continuous action, it is a closed fibre product and hence Polish; the operations are continuous. The equality condition defining composable pairs is likewise closed. These are the full hypotheses of the earlier semidirect-product Theorem 3.1. Apply this to the one-object group \(\operatorname{Aut}(M)\) acting on the cocycle groupoid by \[ \theta(a,\alpha)=(\theta(a),\theta\alpha\theta^{-1}). \] Automorphisms preserve the cocycle identity, products, units and inverses, so this is a groupoid action. With arrow \((\theta,a,\alpha)\), formulas (GEX.11)–(GEX.12) become \[ \begin{aligned} s(\theta,a,\alpha)&=\alpha,& r(\theta,a,\alpha)&=\theta\alpha^a\theta^{-1},\\ (\theta_2,b,\beta)(\theta_1,a,\alpha) &=(\theta_2\theta_1,\theta_1^{-1}(b)a,\alpha),\\ \beta&=\theta_1\alpha^a\theta_1^{-1}. \end{aligned} \tag{GEX.13} \] The inverse is \((\theta^{-1},\theta(a^*),\theta\alpha^a\theta^{-1})\). The printed source \(\theta\alpha\theta^{-1}\) is inconsistent with this product; it must be \(\alpha\). For a concrete test take \(G=\mathbb Z/2\), \(M=M_3(\mathbb C)\), let the nontrivial action be conjugation by the permutation \((01)\), and let \(\theta\) be conjugation by \((12)\). With \(a=1\), the actual arrow goes from \(\alpha\) to the distinct action conjugating by \((02)\). Its product with the unit at \(\alpha\) is allowed by (GEX.13), whereas the printed source would forbid it. Finally an ordinary action of \(G\) on \(X\) produces \(G\ltimes X\) by acting on the **unit groupoid** of \(X\). Its source is \(x\), range is \(gx\), and \((g_1,g_2x)(g_2,x)=(g_1g_2,x)\). The pair groupoid referenced in the printed example is different. Acting on \(X\times X\) gives arrows \((g,x,y)\), source \(y\), range \(gx\), and \[ (g_1,x_1,y_1)(g_2,x_2,y_2) =(g_1g_2,g_2^{-1}x_1,y_2) \] when \(y_1=g_2x_2\). For each \(y\), every \(g\) gives the isotropy arrow \((g,g^{-1}y,y)\). This is generally not a transformation groupoid: even the trivial group on a two-point space gives four pair arrows instead of two unit arrows. ## 6. Localization and uniform cohomology Let \(M=N\bar\otimes L^\infty(X,\mu)\), with \(N\) a separable factor and \(\mu\) standard sigma-finite, and let a separable locally compact group \(G\) act continuously on \(M\). Write its centre action as \((\alpha_gf)(T_gx)=f(x)\). The earlier localization Theorem 5.2 and ANC-01–02 give a strict measurable field action on the full labelled groupoid: \[ (\alpha_ga)(T_gx)=\Theta(g,x)(a(x)),\qquad \Theta(gh,x)=\Theta(g,T_hx)\Theta(h,x). \tag{GEX.14} \] This retains the original \(G\), including its isotropy. It uses the standard forms and Haar repair proved in those lessons and does not require \(G\) itself to be second countable. If \(a_g\in\mathcal U(M)\) is an \(\alpha\)-cocycle, define its localized value at the range by \[ \bar a(g,x)=a_g(T_gx). \] The original identity \(a_{gh}=a_g\alpha_g(a_h)\) becomes \[ \bar a(gh,x)=\bar a(g,T_hx)\, \Theta(g,T_hx)(\bar a(h,x)). \tag{GEX.15} \] Theorem 6.1 of the localization lesson proves that strict Borel representatives with (GEX.15) exist on one invariant conull set, agreeing with each original \(a_g\) separately almost everywhere. Conversely a strict field with that identity gives, by evaluation at \(T_g^{-1}y\), an algebra-valued cocycle. Its coefficients are Borel by field integration; the automatic-continuity Lemma 4.2 makes it continuous. These exact earlier proofs supply both directions of part 7(b). For part 7(c), the group cohomology equation is \[ b_g=f^*a_g\alpha_g(f),\qquad f\in\mathcal U(M). \] It is equivalent to the existence of a Borel unitary field \(F\), on one invariant conull source set, satisfying **for all** \(g,x\) there \[ \bar b(g,x)=F(T_gx)^*\bar a(g,x)\Theta(g,x)(F(x)). \tag{GEX.16} \] The nontrivial point is the common source set. With strict \(\bar a,\bar b,\Theta\), define the action on the unitary fibres \[ Q(g,x)v=\bar a(g,x)\Theta(g,x)(v)\bar b(g,x)^*. \] Using (GEX.15) twice gives \(Q(gh,x)=Q(g,T_hx)Q(h,x)\) exactly. The original group equation says \(f(T_gx)=Q(g,x)f(x)\) for each fixed \(g\) almost everywhere. The Haar-equivariance repair, localization Theorem 1.2, gives \(F=f\) almost everywhere with this equation for every \(g\) on one invariant conull set. Rearranging gives (GEX.16). Conversely (GEX.16), evaluated at \(y=T_gx\), gives the group equation in \(M\) by nonsingularity. This is the complete uniform-cohomology argument of localization Theorem 7.1. An uncountable intersection of fixed-\(g\) null exceptions would not prove it. ## 7. Abelian stabilizers and the product groupoid Let a countable abelian \(G\) act ergodically and nonsingularly on \(X\), and let \(H\) be its kernel on \(L^\infty(X)\). For each \(g\), the Borel set \(\operatorname{Fix}(g)\) is invariant: commutativity gives \(T_gT_kx=T_kT_gx\). Its measure is therefore null or conull. Also \(g\in H\) exactly when its fixed-point set is conull. To see the reverse implication implicit here, a countable separating family of Borel functions on \(X\) is fixed as measurable classes by a kernel element; outside their countably many exceptions those functions separate \(T_gx\) from \(x\), forcing equality. Intersect the fixed-point sets for \(g\in H\) and their complements for \(g\notin H\). Countability gives one invariant conull Borel set \(X_0\) on which \[ G_x=H\quad(x\in X_0). \] Thus \(K=G/H\) acts freely there, with the same principal orbit relation \(R\). The relation is hyperfinite. Here is the amenability input explicitly. Enumerate \(G\) by \(g_1,g_2,\ldots\), and push the uniform measure of \([-n^2,n^2]^n\cap\mathbb Z^n\) to \(G\) by \((k_i)\mapsto\prod_{i=1}^n g_i^{k_i}\). Translation by any fixed \(g_j\), once \(n\geq j\), changes its total \(\ell^1\) mass by at most \(2/(2n^2+1)\). Pushforward cannot increase this distance. These are Reiter probabilities, so the proved invariant-mean construction, Proposition 2.1 and Theorem 6.1 gives hyperfiniteness of the principal measured relation on a conull reduction. ANC-03 now supplies a Borel coherent lift \(c(y,x)\in G\) for every relation arrow, with \[ T_{c(y,x)}x=y,\qquad c(z,y)c(y,x)=c(z,x),\qquad c(x,x)=1. \] Its finite-stage construction includes finite periodic classes and uses no global orbit transversal. Define \[ \begin{aligned} F:G\ltimes X_0&\longrightarrow H\times R,\\ F(g,x)&=(g\,c(T_gx,x)^{-1},T_gx,x),\\ F^{-1}(h,y,x)&=(h\,c(y,x),x). \end{aligned} \tag{GEX.17} \] The first coordinate belongs to the point stabilizer \(H\). Both maps are Borel and inverse to one another. If two arrows have coordinates \((k,z,y)\) and \((h,y,x)\), their original group labels multiply as \[ k c(z,y)h c(y,x)=kh c(z,y)c(y,x)=kh c(z,x). \] Hence (GEX.17) carries their product to \((kh,z,x)\), exactly the product of \(H\times R\). It fixes the units and bijects each counting fibre, so is a measured groupoid isomorphism. This proves both subparts in the full countable abelian scope. There need not be a splitting of the groups. Let \(\mathbb Z\) act cyclically on three points; \(H=3\mathbb Z\). For points represented by \(i,j\in\{0,1,2\}\), take \(c(j,i)=j-i\). Then the first coordinate of (GEX.17) is \(n-(j-i)\in3\mathbb Z\). The product telescopes exactly. But \(\mathbb Z\) cannot be \(3\mathbb Z\times\mathbb Z/3\mathbb Z\), which contains an element of order three. ## 8. Asymptotic ranges and modular endpoints Let \(R\) be a countable nonsingular principal relation and \(\rho:R\to L\) a Borel homomorphism, with \(L\) a locally compact group. Write \(L^+=L\cup\{\infty\}\) for its one-point compactification, with \(\infty\) isolated if \(L\) is compact. Define \[ r_\infty(\rho)= \bigcap_{\mu(E)>0}\operatorname{ess\,ran}_{L^+}(\rho|_{R|E}),\qquad \rho^f(y,x)=f(y)^{-1}\rho(y,x)f(x). \tag{GEX.18} \] Essential ranges use arrow counting measure, not just values on a selected graph. **The nonabelian assertion is false.** A constant cochain \(f=g\) gives \[ r_\infty(\rho^g)=g^{-1}r_\infty(\rho)g. \tag{GEX.19} \] Indeed conjugation extends to a homeomorphism of \(L^+\) fixing \(\infty\); it carries every neighbourhood essential-range test to the corresponding test. The following example shows the resulting set can change. Take the fair Bernoulli space \(X=\{0,1\}^{\mathbb Z}\), shift \((Tx)_k=x_{k+1}\), on its invariant conull aperiodic part. The removed periodic sequences form a countable null set. Define \[ p(n,x)=\sum_{j=0}^{n-1}x_j\pmod2\quad(n>0),\qquad p(-n,x)=-p(n,T^{-n}x). \] The skew transformation \(S(x,i)=(Tx,i+x_0)\) gives \(S^n(x,i)=(T^nx,i+p(n,x))\), proving the cocycle law for all integers. It preserves product measure with the uniform measure on \(\mathbb Z/2\). It is mixing. For cylinder sets \(A,B\) depending on coordinates \([-m,m]\), and fixed fibre values \(i,j\), take \(n>2m+1\). The two cylinder coordinate blocks are disjoint. Between them there is a fair bit, for example \(x_{m+1}\), that occurs once in \(p(n,x)\) and is independent of those blocks. Conditional on all the other relevant bits it makes each parity have probability \(1/2\). Thus \[ (\mu\times u_2)\big((A\times\{i\})\cap S^{-n}(B\times\{j\})\big) =\tfrac14\mu(A)\mu(B). \] Cylinder approximation in measure, with invariance controlling both intersection errors, extends this to arbitrary measurable sets. In particular, for every positive-measure \(E\) and \(d\in\{0,1\}\), \[ \mu\{x\in E:T^nx\in E,\ p(n,x)=d\} \longrightarrow\tfrac12\mu(E)^2>0. \tag{GEX.20} \] This also proves ergodicity of the base shift. Let \(L=S_3\times\mathbb Z\), \(a=((12),0)\), \(g=((123),0)\), and define \(\rho(T^nx,x)=a^{p(n,x)}\). Freeness makes this well-defined; the cocycle law proves it is a homomorphism. Every positive reduction has both values \(e,a\) on positive arrow measure by (GEX.20). It has no other finite value, and its finite image excludes \(\infty\). Hence \[ r_\infty(\rho)=\{e,((12),0)\},\qquad r_\infty(\rho^g)=\{e,((13),0)\}. \tag{GEX.21} \] Both groupoids and target meet the source's separability and ergodicity conditions. This is a counterexample to the full equality in Exercise 9, not a loss caused by a nonmeasurable cochain. **Valid invariants.** Suppose \(L\) is sigma-compact and locally compact; in particular this includes every separable locally compact group under the standing convention. Then for any measurable cochain for which (GEX.18) is defined, \[ r_\infty(\rho)\cap Z(L)=r_\infty(\rho^f)\cap Z(L), \qquad \infty\in r_\infty(\rho)\ \Longleftrightarrow\ \infty\in r_\infty(\rho^f). \tag{GEX.22} \] To check the claimed scope, a separable locally compact group is sigma-compact: take a relatively compact identity neighbourhood \(V\) and countable dense \(D\). Every \(xV^{-1}\) meets \(D\), so \(DV=L\); the compact sets \(d\overline V\) cover \(L\). Here is a proof of (GEX.22) that requires no second-countability restriction. Fix a central \(z\in r_\infty(\rho)\) and a neighbourhood \(U\) of \(z\). For every \(h\in L\), continuity and \(h^{-1}zh=z\) give open neighbourhoods \(W_h\) of \(h\), \(V_h\) of \(z\), with \(W_h^{-1}V_hW_h\subset U\). Sigma-compactness gives a countable subcover of the \(W_h\): take finite subcovers of each compact in a countable compact cover. Given \(\mu(E)>0\), one set \(B=E\cap f^{-1}(W_h)\) has positive measure. Since \(z\) is in every reduced essential range, a positive-measure arrow set of \(R|B\) has \(\rho\in V_h\), and its transformed values lie in \(U\). This proves \(z\in r_\infty(\rho^f)\). Apply the same argument to cochain \(f^{-1}\) and cocycle \(\rho^f\) for the reverse implication. For infinity, let \(K\subset L\) be compact. A countable cover by relatively compact open \(W\)'s supplies a positive \(B=E\cap f^{-1}(W)\). The compact set \(C=\overline W K\overline W^{-1}\) has the property that \(\rho^f(y,x)\in K\) implies \(\rho(y,x)\in C\) for endpoints in \(B\). The positive arrow set with \(\rho\notin C\) therefore has \(\rho^f\notin K\). Reverse the gauge for the converse. In the abelian case every finite point is central, so (GEX.22) proves the full equality originally claimed. The earlier asymptotic-range lesson, §§1–3, proves the same invariants at second countable scope and the counterexample; the countable-cover argument above retains the full sigma-compact scope. **Application to weights.** For \(\nu=q\mu\), \(00}\); hence its asymptotic range in that group's one-point compactification is invariant under equivalent changes of measure. One must distinguish that compactification from \([0,\infty]\). In the one-point compactification of \(\mathbb R_{>0}\), escape towards zero and towards unbounded values reaches the **same** point. For the two endpoints used separately in modular ratio calculations, invariance can be proved directly. Given \(E\) of positive measure, some \[ B=E\cap\{1/n\leq q\leq n\} \] has positive measure. On \(R|B\) the factor \(q(y)/q(x)\) lies between \(n^{-2}\) and \(n^2\). Thus \(\delta_\mu<\varepsilon/n^2\) implies \(\delta_\nu<\varepsilon\), and \(\delta_\mu>n^2K\) implies \(\delta_\nu>K\), each on positive arrow measure when the relevant endpoint belongs to every reduced essential range. Replacing \(q\) by \(1/q\) proves both converses. This invariance does not identify the operators or their domains. In the source-counting Hilbert space, RWT-02 gives \[ \Delta_\nu\xi=\delta_\nu\xi,\qquad D(\Delta_\nu^z)= \left\{\xi:\int_R\delta_\nu^{\,2\operatorname{Re}z}|\xi|^2\,d\nu_s<\infty\right\}. \tag{GEX.24} \] The formula for every complex \(z\) follows from the spectral calculus of a positive multiplication operator: truncate to \(1/k\leq\delta_\nu\leq k\), apply bounded functional calculus there, and let the projections increase; the integrability condition is exactly the domain of the closed limit. Although the real flow preserves the weight's null-set class, an unbounded gauge in (GEX.23) can change this integrability condition. The endpoint invariant (GEX.22) therefore complements the complete operator-domain formula; it does not replace it. ## References M. Takesaki, *Theory of Operator Algebras III*, XIII §3, Exercises 1–9, pp. 56–60. The preceding sections give independently written solutions, including the periodic qualifications, topological hypothesis and corrections to the printed formulas. The Weyl factor is the one specified in XIII §1, Exercise 1. The programme proof links in the introduction and at each application locate the full results used here: the small cochain extension, the all-orbit two-gap construction, the measured suspension passage, normal rotation-factor identification, finite matrix-field stages, strict localization, invariant means and the relation-weight model.