The common conventions and lattice class

For example, \(\mathrm{SL}_3(\mathbb Z)\) and every finite-index subgroup belong to \(\mathscr K\). Section 5 verifies the lattice and ICC hypotheses for these and other named examples.

Property \((T)\) means that every continuous unitary representation with almost invariant unit vectors has a nonzero invariant vector. Standard structure and permanence results for this class are recalled in Section 2; see also (Bekka et al. 2008).

All scalar cocycles take values in \(\mathbb T\) and are normalized: \[\mu(g,h)\mu(gh,j)=\mu(h,j)\mu(g,hj),\qquad \mu(e,g)=\mu(g,e)=1.\] The twisted group factor \(L_\mu(\Gamma)\) has canonical unitaries \(u_g\) with \(u_gu_h=\mu(g,h)u_{gh}\) and trace \(\tau(u_g)=\mathbf1_{\{e\}}(g)\). For an ICC group it is a factor; Section 2 recalls the short Fourier argument.

A bifinite \(M\)–\(N\) correspondence is a Hilbert space with commuting normal unital left \(M\) and right \(N\) actions and finite von Neumann module dimensions on both sides. The actions need not be mutual commutants, and the correspondence need not be irreducible. Every correspondence in this paper is separable; every factor isomorphism is normal, unital, complex-linear, and \(*\)-preserving. For a finite factor \(Q\), write \[Q^t=p\bigl(M_n(\mathbb C)\,\overline\otimes\,Q\bigr)p, \qquad (\mathop{\mathrm{Tr}}_n\otimes\tau_Q)(p)=t>0.\] The matrix trace \(\mathop{\mathrm{Tr}}_n\) is unnormalized; the corner itself has its normalized trace. The fundamental group of \(Q\) is \(\mathcal F(Q)=\{t>0:Q^t\cong Q\}\).

Stable canonical recovery

A stable comparison is an isomorphism between positive amplifications of two factors. The following theorem forces the amplification scales to agree and determines the given map on its actual matrix corners.

The untwisted specialization answers the stable recovery question for Definition 1, with an arbitrary ICC comparison group. Property \((T)\) alone does not imply this recovery. OpenAI constructed a countably infinite family of pairwise nonisomorphic ICC property-\((T)\) groups with isomorphic group factors (OpenAI 2026c, chap. 4, Theorem 1.2). Independently and concurrently, Zhou constructed another pair (Zhou 2026, Theorem A).

Arbitrary finite-index factor neighbors

We next allow the second factor to have no given group presentation. A finite-index factor neighbor of \(M\) is a factor joined to \(M\) by a nonzero bifinite correspondence. The arithmetic data describing such a neighbor involve finite-index subgroup isomorphisms rather than an isomorphism of the whole groups.

For a group \(A\), a normalized projective unitary representation with multiplier \(\beta\) is a map \(\sigma:A\to\mathcal U(V)\) satisfying \(\sigma(a)\sigma(b)=\beta(a,b)\sigma(ab)\) and \(\sigma(e)=I\). We call \(\beta\) finite type when such a representation exists with \(0<\dim_{\mathbb C}V<\infty\); its image need not be finite. For a group homomorphism \(\delta\), the pullback cocycle is \(\delta^*\omega(a,b)=\omega(\delta(a),\delta(b))\).

The unknown factor \(P\) is not assumed to have a group presentation or a preferred basis. Its presentation is recovered from multiplication in a finite-index extension. If \(\mu=1\), finite-index projective induction recovers the usual finite-type condition on \(\omega\). For arbitrary \(\mu\) the ratio condition is essential: \(P=L_\mu(\Gamma)\) is always an endpoint.

The arithmetic correspondence category

The two recovery theorems use a common classification of correspondences. We first describe its models and then the finite-dimensional data that control their maps and summands.

For finite-index subgroups \(A\leq\Gamma\) and \(B\leq\Lambda\), an isomorphism \(\delta:A\to B\), and a nonzero finite-dimensional projective unitary representation \(\sigma\) satisfying \[ \sigma(a)\sigma(b) =\frac{\mu(a,b)}{\omega(\delta(a),\delta(b))}\sigma(ab), \qquad \sigma(e)=I, \tag{2}\] define \[ E(A,B,\delta,\sigma) =L^2(M)\,\overline\otimes_{L_{\mu|_A}(A)} \bigl(V_\sigma\otimes L^2(N)\bigr), \qquad M=L_\mu(\Gamma),\quad N=L_\omega(\Lambda). \tag{3}\] The right \(N\) action is standard. The left subgroup action on the second factor is \(u_a(\xi\otimes\eta)=\sigma(a)\xi\otimes v_{\delta(a)}\eta\). Equation (2) makes this an action with multiplier \(\mu|_A\).

A right action is represented by \(\rho(h)\xi=\xi v_h^*\), so \(\rho(h)\rho(k)=\overline{\omega(h,k)}\rho(hk)\). The left action is \(\pi(g)\xi=u_g\xi\). Together they give the joint action \(U_{(g,h)}\xi=u_g\xi v_h^*\).

A virtual-isomorphism germ from \(\Gamma\) to \(\Lambda\) is an isomorphism between finite-index subgroups, modulo agreement on a common finite-index subgroup. The action of \(\Gamma\times\Lambda\) on germs is \[(g,h)\cdot[\delta] =[\mathop{\mathrm{Ad}}(h)\circ\delta\circ\mathop{\mathrm{Ad}}(g^{-1})].\] For a germ \(d\) let \(C_d\) denote its full stabilizer. ICC implies that \(C_d\) is the graph of an isomorphism \(A_d\to B_d\) between finite-index subgroups; the proof is included in Section 3. Use the joint multiplier \[ \alpha\bigl((g,h),(g',h')\bigr) =\mu(g,g')\overline{\omega(h,h')}. \tag{4}\] For a finite-dimensional \(\alpha|_{C_d}\)-projective representation \(\sigma_d\), define \[H(d,\sigma_d)=E(A_d,B_d,\delta_d,\widetilde\sigma_d), \qquad \widetilde\sigma_d(a)=\sigma_d(a,\delta_d(a)).\] Here \(\delta_d:A_d\to B_d\) is the isomorphism whose graph is \(C_d\). Section 3 realizes \(H(d,\sigma_d)\) as the orthogonal sum of copies of the representation space indexed by \((\Gamma\times\Lambda)/C_d\), and describes the induced transports between them.

The full stabilizer matters because a graph presentation can split one of these fibres into several pieces. If \(C_\delta=\{(a,\delta(a)):a\in A\}\) represents \(d\), the finite set \(C_d/C_\delta\) indexes the pieces that must be grouped together. Their projective representation is induced from \(C_\delta\) to \(C_d\) with multiplier \(\alpha\). Bimodule projections may mix these pieces; they become fibre projections only after this grouping.

The integer dimensions in part (ii) are the dimensions of the actual correspondence, including a proper summand of a presenting model. That distinction is essential to the stable recovery argument: the two dimensions of an equivalence correspondence are reciprocal, so here both must equal one. The projective data throughout concern only the matched ratio in Equation (2); neither individual endpoint cocycle is assumed finite type.

Theorems 2 and 3 will follow from this map-level classification in Section 4. These are structural classifications; they do not assert a decision procedure for arbitrary group presentations or cocycle classes.

For untwisted factors, the same results identify the outer automorphism group and recover the group from its reduced C*-algebra whenever that algebra has a unique tracial state. Section 5 also gives commensurability and finite-type neighbor criteria, named lattice examples, and the finite-index endomorphism consequence for \(L(\mathrm{SL}_3(\mathbb Z))\). The last consequence uses a separate classical covolume argument.

History and the role of the arithmetic category

Popa’s deformation/rigidity theory produced rigidity theorems for crossed products and Bernoulli actions. Ioana’s \(W^*\)-superrigidity theorem for Bernoulli actions of ICC property-\((T)\) groups concerns an action together with its crossed product (Ioana 2011). Ioana–Popa–Vaes obtained the first \(W^*\)-superrigid groups, in classes of generalized wreath products (Ioana et al. 2013). Chifan–Ioana–Osin–Sun constructed the first such groups with property \((T)\), using wreath-like products (Chifan et al. 2023). Rigidity of an action and recovery from a bare group factor are distinct problems, and the correspondence category asks additionally for finite index, multiplicities, and composition.

The geometric route for lattices has its own antecedents. For lattices in \(\mathop{\mathrm{Sp}}(n,1)/\{\pm1\}\), \(n\geq2\), Cowling–Haagerup’s completely bounded approximation invariant already determines \(n\) from the group factor (Cowling and Haagerup 1989, Main theorem and Propositions 6.1–6.2). Bekka classified the characters of \(\mathrm{SL}_n(\mathbb Z)\) for \(n\geq3\), and Peterson proved a finite-factor representation dichotomy for irreducible higher-rank property-\((T)\) lattices (Bekka 2007; Peterson 2014). Boutonnet–Houdayer developed stationary and noncommutative boundary methods and recovered character rigidity in that framework (Boutonnet and Houdayer 2021). These results provide important context for the companion’s geometry. They do not by themselves identify all finite correspondences to arbitrary ICC groups. The companion proves the exhaustion input in the full mixed-local-field class stated here, with reducible products and arbitrary scalar twists.

Jones’s index and the expectation inequality of Pimsner–Popa provide the dimension and positivity framework (Jones 1983; Pimsner and Popa 1986). We use Connes’s correspondence category and relative tensor product (Connes 1994, V, Appendix B). Complete finite-index classifications and explicit fusion rules already occur in Vaes’s work on generalized Bernoulli crossed products (Vaes 2008, Theorem 2.2 and Section 6.2). Theorem 2.2 allows scalar twists at both endpoints and a finite-dimensional projective representation of their matched ratio; Definition 6.8 and Theorem 6.9 supply projective induction and the cocycle-corrected double-coset rule. We adapt this calculus to graph bundles for twisted group factors. The full stabilizer of a virtual-isomorphism germ identifies every bounded intertwiner and every selecting projection, including projections that mix smaller presenting fibres.

Donvil–Vaes prove rigidity under arbitrary scalar twists and virtual isomorphism for certain left-right wreath products (Donvil and Vaes 2025). Their Proposition 6.5 reconstructs an intermediate factor from finite-dimensional homogeneous spaces and their actual products, and Theorem 6.6 applies this construction to arbitrary bifinite factor neighbors. The source cocycle is already arbitrary there, and the conclusion retains a finite-type ratio of the two cocycles. We adapt their homogeneous-product and minimal-corner argument from automorphism degrees to a grading by virtual automorphism germs of \(\Gamma\); finite-index domains and boundedness of the homogeneous vectors are treated explicitly. The companion supplies the distinct exhaustion theorem for the local-field lattice class. The finite-type ascent used for an untwisted source is the projective-induction statement of (Donvil and Vaes 2025, Lemmas 6.7–6.8).

Chifan–Fernández Quero–Osin–Tan obtain stable canonical recovery for scalar twists of certain property-\((T)\) wreath-like products when the source cocycle is trivial on the abelian base (Chifan et al. 2026, Theorem F). Their comparison group is arbitrary ICC. The unrestricted source cocycles and lattice class treated here enter through the companion’s distinct exhaustion theorem.

Outside the property-\((T)\) setting, the nonisomorphic free groups \(\mathbb F_2\) and \(\mathbb F_3\) satisfy \(L(\mathbb F_2)\cong L(\mathbb F_3)\) (OpenAI 2026a, Theorem 1.2). These factors have fundamental group \(\mathbb R_{>0}\) (OpenAI 2026a, Corollary 6.1), in contrast to the trivial fundamental groups in Theorem 2.

Proof strategy and navigation

There are three algebraic steps. First, the graph models admit orthogonal finite-dimensional fibres indexed by graph cosets. Fibres with the same virtual-isomorphism germ must be grouped together before passing to a summand. The full stabilizer of a germ has only one finite orbit on the germ space. Square summability of each column of a bounded intertwiner therefore forces that map to preserve each grouped fibre. This proves the Hom formula and makes the two dimensions sums of integer fibre ranks times subgroup indices.

Second, a bifinite correspondence with an unknown factor produces a finite-index extension of the known group factor. Arithmetic exhaustion grades its \(L^2\) space by finite-dimensional germ fibres. The Pimsner–Popa inequality turns those vectors into bounded operators. Products can then be computed on a common subgroup domain, and they multiply the germ degrees. In a minimal corner of the identity fibre, every nonzero degree is a unitary line. Those lines, with their actual products and trace, recover the group and scalar cocycle. The identity block supplies the projective representation of the cocycle ratio.

Third, the equivalence correspondence of a specified stable isomorphism has reciprocal dimensions. Their integrality forces both to be one, leaving one full graph and one one-dimensional projective fibre. The correspondence identification recovers the cochain. The column realizations in the given matrix corners then recover the implementing partial isometry for the specified map. Figure 1 separates these two consequences of the category calculation.

Section 2 records the relative tensor convention and exact companion interface. Section 3 proves all model formulas, then applies exhaustion. Section 4 reconstructs factor neighbors and specified stable maps. Section 5 proves the further consequences described above.

Relative tensor products and elementary dimensions

All Hilbert-space inner products are linear in the second variable. A vector \(\xi\) in an \(M\)–\(N\) correspondence is right bounded if \(R_\xi\widehat a=\xi a\) extends to a bounded map from \(L^2(N)\). For an \(N\)–\(P\) correspondence, the relative tensor product is the completion, after quotienting out zero-norm vectors, of the tensors with right-bounded first vector and inner product \[ \langle\xi\otimes_N\eta,\xi'\otimes_N\eta'\rangle =\langle\eta,(R_\xi^*R_{\xi'})\eta'\rangle. \tag{6}\] The right-module map \(R_\xi^*R_{\xi'}\) is left multiplication by an element of \(N\). Balancing means \((\xi a)\otimes_N\eta=\xi\otimes_N(a\eta)\) for \(a\in N\). This is the finite tracial form of Connes’s relative tensor product (Connes 1994, V, Appendix B, Theorems 13 and 20).

A finite right \(N\)-module has a column realization \(pL^2(N)^n\), with dimension \((\mathop{\mathrm{Tr}}_n\otimes\tau_N)(p)\). The vectors in \(pN^n\) are right bounded and dense. The projection model and its dimension are independent of the chosen realization (Jones 2009, Theorem 10.1.1 and Proposition 10.1.3). We prove the normality of the graph models directly in Lemmas 7 and 8. The same elementary construction precedes the geometric argument in the companion, so its exhaustion proof has no dependence on our all-maps or summand classification. For \(n=\dim\sigma\), the elementary dimensions are \(n[\Lambda:B]\) and \(n[\Gamma:A]\) in left/right order. The dimensions of actual summands will require the full-stabilizer calculation.

The geometric input

The real simple factors in Definition 1 have higher rank or are locally \(\mathop{\mathrm{Sp}}(n,1)\), \(n\geq2\), or \(F_{4(-20)}\). At a finite place their relative rank is at least two. Complex places are treated as real Lie groups in differential arguments. Property \((T)\) passes between a locally compact group and a lattice and between a discrete group and a finite-index subgroup; see (Bekka et al. 2008). In particular, that class is closed under abstract commensurability among countably infinite ICC groups. The geometric companion treats arbitrary lattices in the permitted finite products, without a dense-projection hypothesis. We use exactly the following theorem.

This is the Arithmetic exhaustion theorem of the complete companion (OpenAI 2026b, Theorem 1.2). Its conclusion retains actual finite-index subgroup isomorphisms and their exact scalar ratios. We do not assume that the projective representations have finite image, that either cocycle is finite type, or that the original correspondence is irreducible. The algebraic proofs that follow are independent of this proposition until its explicit application at the end of Section 3.

Germs and their full stabilizers

A germ from \(\Gamma\) to \(\Lambda\) is an isomorphism \(\delta:A\to B\) between finite-index subgroups, modulo agreement after restriction to a common finite-index subgroup. Composition means composition on a finite-index domain where both representatives are defined. For example, if \(\varepsilon:C\to F\) is a representative from \(\Lambda\) to a third group, then \(\varepsilon\delta\) is defined on \(\delta^{-1}(B\cap C)\) and has image \(\varepsilon(B\cap C)\). Both have finite index in the relevant endpoint group. Restricting representatives further does not change the germ, and a common further restriction verifies associativity. Inversion is represented by the inverse subgroup isomorphism.

For a germ \(d\) from \(\Gamma\) to \(\Lambda\), let \[(g,h)d=\mathop{\mathrm{Ad}}(h)\,d\,\mathop{\mathrm{Ad}}(g^{-1}),\qquad C_d=\{(g,h)\in D:(g,h)d=d\}.\] If \(\delta:A\to B\) represents \(d\), denote its graph by \(C_\delta=\{(a,\delta(a)):a\in A\}\). We distinguish this graph from the full stabilizer \(C_d\).

The finite-orbit conclusion will make germ labels intrinsic to the induced models. Cosets of \(C_d\) correspond to actual germs, whereas several cosets of a presenting graph may represent the same germ. After grouping those cosets, a bounded intertwiner will have matrix blocks between the resulting finite-dimensional fibres. Equivariance makes a block’s norm constant along a stabilizer orbit. The lemma will therefore force such a map to preserve each germ label, once square summability of its columns is used.

Projective induction and the arithmetic model

Projective induction, induction in stages, and double-coset tensor decompositions belong to Mackey’s representation-theoretic framework (Mackey 1958, Theorems 4.1, 4.2, and 4.6). The following coordinates fix the multiplier throughout and prove every scalar identity used here.

Let \(C\leq D\) be a subgroup and let \(\gamma\) be a normalized unitary projective representation on a finite-dimensional space \(V\), with multiplier \(\alpha|_{C\times C}\). Write \(X=D/C\) for left cosets. Choose representatives \(r_x\in D\), with \(r_C=e\), and give each \(x\in X\) a copy \(V_x\) of \(V\). Set \(\mathcal H(C,\gamma)=\bigoplus_{x\in X}V_x\), with the counting Hilbert norm. For \(s\in D\) put \[ \begin{split} c(s,x)&=r_{sx}^{-1}sr_x,\\ U^X_s(x)&= \frac{\alpha(s,r_x)}{\alpha(r_{sx},c(s,x))}\, \gamma(c(s,x)):V_x\longrightarrow V_{sx}. \end{split} \tag{10}\]

We next identify this construction with the Connes-product model in Theorem 4. This also supplies a direct normality check of the subgroup action used to define that product.

For a full stabilizer we write \[H(d,\gamma)=\mathcal H(C_d,\gamma).\] Its labels identify with actual germs in the orbit \(Dd\). There is also a canonical regrouping of an original graph model over this full-germ orbit. The map \(D/C_\delta\to D/C_d\) has finite fibres. Its fibre above \(C_d\) carries \[ \operatorname{Ind}^{C_d,\alpha}_{C_\delta}\gamma, \tag{14}\] where projective induction uses Equation (10) with \(D\) replaced by \(C_d\) and with the restricted cocycle. Indeed that fibre is the sum indexed by \(C_d/C_\delta\), and its transports are precisely that formula. Regrouping the orthogonal sum consequently gives \[\mathcal H(C_\delta,\gamma) \cong H\bigl(d,\operatorname{Ind}^{C_d,\alpha}_{C_\delta}\gamma\bigr).\] This description includes the phases of finite induction; it does not replace the projective representation by an ordinary one.

Changing an orbit representative also retains a scalar transport. If \(d'=kd\), then \(C_{d'}=kC_dk^{-1}\). Identify the new base fibre with the old one using \(U_k\). For \(c'=kck^{-1}\in C_{d'}\) the representation in these coordinates is \[ \gamma'(c')= \frac{\alpha(c',k)}{\alpha(k,c)}\,\gamma(c). \tag{15}\] Indeed \(U_{c'}U_k= \alpha(c',k)\alpha(k,c)^{-1}U_kU_c\). This gives the exact representative-change equivalence, including its restricted multiplier. Two choices of \(k\) leading to the same germ give equivalent fibres by the stabilizer action itself.

All bounded intertwiners and all summands

Consider any closed bimodule summand \(K\) of a finite sum of the original graph models. Regroup each model by Equation (14), combine copies of the same full-germ orbit, and apply Theorem 10 to the projection onto \(K\). We obtain a finite orthogonal sum of full-germ bundles, with exactly the intertwiners in Equation (17). In particular, if its fibre representations are \(\rho_d\), its actual dimensions are \[ \dim_M K=\sum_d\dim_{\mathbb C}(\rho_d)[\Lambda:B_d], \qquad \dim K_N=\sum_d\dim_{\mathbb C}(\rho_d)[\Gamma:A_d]. \tag{19}\] Both are positive integers for \(K\ne0\). This conclusion uses the description of its projection; the dimensions of a containing model alone would not imply it.

Conjugation

On the conjugate correspondence \(\overline{\mathcal H(C,\gamma)}\) the endpoint groups are swapped. Its subgroup and base fibre are \[C^{\mathrm{op}}=\{(h,g):(g,h)\in C\},\qquad \bar\gamma(h,g)\bar\xi=\overline{\gamma(g,h)\xi}.\] The multiplier is \(\omega\boxtimes\bar\mu\), since \(\bar\gamma\) conjugates \(\alpha\) and swaps its two variables. The label \(x\) of a full-germ bundle becomes the inverse germ \(x^{-1}\): indeed \[\bigl(\mathop{\mathrm{Ad}}(h)x\mathop{\mathrm{Ad}}(g^{-1})\bigr)^{-1} =\mathop{\mathrm{Ad}}(g)x^{-1}\mathop{\mathrm{Ad}}(h^{-1}).\] With swapped coset representatives, the transport is explicitly \[ U^{\bar X}_{(h,g)}(\bar x) =\overline{U^X_{(g,h)}(x)}. \tag{20}\] Thus it includes the conjugate of the scalar ratio in Equation (10). At the module level this equality is the defining identity \(v_h\bar\xi u_g^*=\overline{u_g\xi v_h^*}\). For an original graph \(\delta:A\to B\), the new graph is \(\delta^{-1}:B\to A\) and its fibre representation is \(b\mathrel{% \pdfliteral direct{/Span << /ActualText <FEFF21A6> >> BDC}% \paperoriginalmapsto \pdfliteral direct{EMC}}\overline{\sigma(\delta^{-1}(b))}\). Fibre intertwiners are conjugated by the same rule; their adjoints and norms are preserved. Proposition 9 also shows directly that the left and right dimensions are exchanged.

Balanced fusion and its scalar transport

Let \(P=L_\nu(\Pi)\). Consider a bundle \(\mathcal H_X\) from \((\Gamma,\mu)\) to \((\Lambda,\omega)\) and a bundle \(\mathcal H_Y\) from \((\Lambda,\omega)\) to \((\Pi,\nu)\), each constructed as above. Write \(V_x\) and \(W_y\) for their fibres and \(\alpha'=\omega\boxtimes\bar\nu\). Finite direct sums are handled by distribution, so we may first take one transitive bundle at each endpoint. Define \[ Z=(X\times Y)/\Lambda, \qquad h(x,y)=\bigl((1,h)x,(h,1)y\bigr). \tag{21}\] Identify the tensor fibres along this action using \[ J_h(x,y)=U^X_{(1,h)}(x)\otimes U^Y_{(h,1)}(y). \tag{22}\] These identifications are genuine unitary group transports, because their multiplier is \(\overline{\omega(h_1,h_2)}\omega(h_1,h_2)=1\). The action on labels is free, as already its action on \(X\) is free. Each quotient fibre \(V_z\) is consequently unambiguously a tensor fibre after choosing one representative of \(z\). Put \(\mathcal H_Z=\bigoplus_{z\in Z}V_z\), with counting norm at any such representatives. Changing representatives gives a unitary identification by Equation (22).

The outer transport before passage to the quotient is \(U^X_{(g,1)}\otimes U^Y_{(1,k)}\). It commutes exactly with every \(J_h\): each mixed-coordinate factor of either product cocycle is one. Its multiplier is \(\mu\boxtimes\bar\nu\). For an explicit formula choose a representative \((x,y)\) of \(z\) and representatives for all quotient labels. There is a unique \(h\) such that \[\bigl((g,h)x,(h,k)y\bigr)\] is the chosen representative of \((g,k)z\). With \(s=(g,h)\), \(t=(h,k)\), \(c=c_X(s,x)\) and \(b=c_Y(t,y)\), the quotient transport is \[ \frac{\alpha(s,r_x)\alpha'(t,r_y)} {\alpha(r_{sx},c)\alpha'(r_{ty},b)} \gamma_X(c)\otimes\gamma_Y(b). \tag{23}\] It is the product of the two transports in Equation (10). There is no additional factor from the compensating element \(h\), since its lift is the genuine transport in Equation (22).

To attach a germ to a quotient label, first recall the map from graph cosets to germs. For \(x=(g,h)C_\delta\in X\), put \[d_x=\mathop{\mathrm{Ad}}(h)[\delta]\mathop{\mathrm{Ad}}(g^{-1}),\] and define \(d_y\) in the same way for \(y\in Y\). These are the graph-coset-to-germ maps from the earlier regrouping, applied at their respective endpoints; they need not be injective. The middle action changes the two associated germs by \[(d_x,d_y)\mathrel{% \pdfliteral direct{/Span << /ActualText <FEFF27FC> >> BDC}% \paperoriginallongmapsto \pdfliteral direct{EMC}} (\mathop{\mathrm{Ad}}(h)d_x,\,d_y\mathop{\mathrm{Ad}}(h^{-1})).\] Hence \(z=[(x,y)]\mathrel{% \pdfliteral direct{/Span << /ActualText <FEFF21A6> >> BDC}% \paperoriginalmapsto \pdfliteral direct{EMC}}d_y d_x\) is well defined. Distinct quotient labels can still have the same germ; the next proposition explains how their fibres are regrouped.

For completeness, the last proof yields an explicit finite double-coset formula without unspecified stabilizer representations. Let \(\sigma\) and \(\tau\) be the original fibre representations on \(A\) and \(C\). In the fibre at \(z_m\) choose \(r_{(m,1)C_\varepsilon}=(m,1)\). For \(a\in A_m\), write \(b=\delta(a)\) and \(c=m^{-1}bm\in C\). Equation (23), with compensating element \(h=b\), gives \[ \gamma_m(a)= q_m(b)\,\sigma(a)\otimes\tau(m^{-1}bm), \qquad q_m(b)=\frac{\omega(b,m)}{\omega(m,m^{-1}bm)}. \tag{29}\] The correction has the exact coboundary \[ \frac{q_m(b)q_m(b')}{q_m(bb')} =\frac{\omega(b,b')} {\omega(m^{-1}bm,m^{-1}b'm)}. \tag{30}\] To verify this, the twisted group-unit identity \(v_bv_m=q_m(b)v_mv_{m^{-1}bm}\) can be multiplied successively for \(b,b'\) and compared with the identity for \(bb'\). Combining Equation (30) with the multipliers of \(\sigma\) and \(\tau\) shows that \(\gamma_m\) has multiplier \(\mu|_{A_m}/\theta_m^*\nu\). Hence \[ E(A,B,\delta,\sigma)\mathbin{\bar\otimes}_N E(C,F,\varepsilon,\tau) \cong\bigoplus_{m\in B\backslash\Lambda/C} E(A_m,F_m,\theta_m,\gamma_m). \tag{31}\] The appropriate endpoint algebras in the second model and in each summand are understood from their displayed groups. One then applies finite projective induction to the full stabilizers and combines any equal full-germ orbits. Formula (29) exhibits explicitly why replacing the twisted fusion rule by its ordinary version would lose a phase.

The indices in Equation (27) satisfy \[[\Gamma:A_m]=[\Gamma:A][B:B\cap mCm^{-1}], \qquad [\Pi:F_m]=[\Pi:F][C:C\cap m^{-1}Bm].\] Counting the left \(B\)-orbits in \(\Lambda/C\) and the right \(C\)-orbits in \(B\backslash\Lambda\) gives, respectively, \[\sum_m[B:B\cap mCm^{-1}]=[\Lambda:C], \qquad \sum_m[C:C\cap m^{-1}Bm]=[\Lambda:B].\] Thus both coordinate groups have finitely many orbits in the fused label space. These formulas also verify, directly from the models and their counting norms, the usual multiplicativity of the left dimensions and of the right dimensions under this fusion.

Maps, associativity, and completion of the calculus

The preceding identifications respect the entire operator calculus, not only the isomorphism classes of objects. For the map calculation, first regroup the inputs into full-germ bundles. Theorem 10 then represents every bounded intertwiner by an equivariant family of fibre maps. An equivariant family \(T_x:V_x\to V'_x\) and an equivariant family \(S_y:W_y\to W'_y\) give the fibre maps \(T_x\otimes S_y\). They commute with the middle identifications in Equation (22), so descend to the quotient fibres. Under the unitary of Proposition 12, they are precisely \(T\mathbin{\bar\otimes}_N S\), as is seen on \(\xi\otimes_N\eta\). Their norms are bounded by \(\|T\|\|S\|\); adjoints and compositions agree fibrewise. Regrouping by finite projective induction is a regrouping of these same Hilbert fibres and maps. Conjugation has the map formula already given in Equation (20).

For three bundles, both orders of fusion use the quotient of the triple label space by the two middle groups. Their actions commute: on the shared middle bundle they are its commuting coordinate actions, and their projective factors cancel with those in the adjacent bundles. The fibre identification between the two orders is the ordinary unitary \[(\xi\otimes\eta)\otimes\zeta \mathrel{% \pdfliteral direct{/Span << /ActualText <FEFF27FC> >> BDC}% \paperoriginallongmapsto \pdfliteral direct{EMC}}\xi\otimes(\eta\otimes\zeta).\] The counting norms and Equation (25) show that it is also the canonical associativity unitary for the relative products. All scalar identifications are those of Equation (22); hence this assertion requires no change of cocycle. For any further number of factors these maps agree on elementary tensors, which are total, so the associativity identifications are coherent. The unit correspondence \(L^2(M)\) is the identity-germ model with its one-dimensional trivial graph representation; its unit maps are ordinary left and right multiplication on the corresponding relative tensors.

Conjugation reverses a balanced product. The unitary \[\overline{\mathcal H_X\mathbin{\bar\otimes}_N\mathcal H_Y} \longrightarrow \overline{\mathcal H_Y}\mathbin{\bar\otimes}_N \overline{\mathcal H_X}, \qquad \overline{\xi\otimes_N\eta} \mathrel{% \pdfliteral direct{/Span << /ActualText <FEFF27FC> >> BDC}% \paperoriginallongmapsto \pdfliteral direct{EMC}}\bar\eta\otimes_N\bar\xi\] is well defined on tensors of vectors in labelled fibres: the simultaneous middle transport becomes the same transport on the swapped conjugate fibres, with both scalar factors conjugated and their product still one. The counting tensor-fibre model of Proposition 12 proves preservation of the norm: the map bijects the quotient labels and is unitary on each conjugate tensor fibre. These tensors are total. It intertwines the swapped endpoint actions and the conjugated fibre maps. Thus the conjugation, unit, and associativity identifications agree with those of the correspondence category at the level of actual operators.

We now apply the geometric input. Proposition 5 places every bifinite correspondence with source in \(\mathscr K\) inside a finite sum of graph models. The preceding full-germ calculation resolves its selecting projection into fibre projections. Together with the operation formulas just proved, this establishes every assertion of Theorem 4, including all bounded intertwiners, actual summands and their dimensions, and the specified conjugation and fusion maps. Throughout the section only the matched-subgroup ratio is required to have a finite-dimensional projective realization. None of the calculations imposes a finite-image condition on that representation or a finite-type condition on either individual endpoint cocycle.

A finite-index extension and its intrinsic grading

Fix \(\Gamma\in\mathscr K\), an arbitrary normalized scalar cocycle \(\mu\), and \(M=L_\mu(\Gamma)\). Write \(D=\mathop{\mathrm{Comm}}(\Gamma)\) for the group of virtual automorphism germs defined in Section 3. The map \[\iota:\Gamma\longrightarrow D,\qquad \iota(g)=[\mathop{\mathrm{Ad}}(g)],\] is injective: an element in its kernel centralizes a finite-index subgroup of \(\Gamma\), and hence has a finite conjugacy class. For a representative \(\delta:A\to B\) of \(d\in D\), one has \[ d\,\iota(a)\,d^{-1}=\iota(\delta(a))\qquad(a\in A). \tag{32}\]

Bounded homogeneous operators and the minimal corner

Let \(M\subset Q\) now be a unital finite-index inclusion of \(\mathrm{II}_1\) factors and apply Lemma 14 to \({}_M L^2(Q)_M\). Until the next lemma, its degree spaces are only Hilbert subspaces of \(L^2(Q)\).

Compressing \(K_1\) to a scalar algebra will now force each surviving degree to be a unitary line.

The ratio condition and the reconstructed group

The minimal corner has supplied a twisted group algebra, but the group \(\Lambda\) still needs to satisfy the hypotheses of Theorem 3. We next recover a finite-index copy of a subgroup of \(\Gamma\) inside \(\Lambda\), obtain the projective ratio from the identity matrix block, and prove that \(\Lambda\) is ICC.

Stable scales and the actual isomorphism

For a specified stable isomorphism, the category gives a different constraint: its equivalence correspondence has reciprocal dimensions. After integrality forces both dimensions to be one, we will use the given corner columns to identify the specified map. All dimensions below use the normalized factor traces and the ordinary unnormalized matrix trace \(\mathop{\mathrm{Tr}}_n\), with \(\mathop{\mathrm{Tr}}_n(1)=n\).

Untwisted correspondences and finite-type neighbors

With the finite-type convention of Section 1, the endpoint theorem specializes as follows.

For arbitrary source twist this specialization must be replaced by the ratio in Theorem 3. In particular, its direct proof admits the identity correspondence of \(L_\mu(\Gamma)\) whatever the individual cocycle \(\mu\) may be.

Examples and specified canonical maps

The familiar arithmetic examples are \(\mathrm{SL}_3(\mathbb Z)\), its principal congruence subgroup \[\Gamma(3)=\ker\bigl(\mathrm{SL}_3(\mathbb Z) \longrightarrow\mathrm{SL}_3(\mathbb Z/3\mathbb Z)\bigr),\] all their finite-index subgroups, \(\mathrm{PSL}_n(\mathbb Z)\) for \(n\geq4\), and \(\mathrm{PSp}_4(\mathbb Z)\), with the latter two symbols denoting the quotient by the center of the integral matrix group. The arithmetic lattice theorem places these projective groups in the corresponding center-free real groups (Borel and Harish-Chandra 1962); the finite-covolume statement appears also in (Borel and Harish-Chandra 1961, Theorem 1(iv)). For even \(n\), \(\mathrm{PSL}_n(\mathbb R)\) is the identity root-generated subgroup of \(\mathrm{PGL}_n(\mathbb R)\); for odd \(n\) it is all of it. Likewise \(\mathrm{PSp}_4(\mathbb R)\) is the root-generated adjoint symplectic group. These ambient groups have real rank at least two and property \((T)\) (Bekka et al. 2008).

These projective lattices are ICC. A finite conjugacy class gives a finite-index centralizer. That subgroup is Zariski dense by Borel density, so the element centralizes the connected ambient algebraic group and is trivial in its adjoint form (Borel 1960). Equivalently, the centralizer formulation in (Furstenberg 1976, Corollary 3) applies directly to the finite-index lattice and gives the ambient center. For \(\mathrm{SL}_3(\mathbb Z)\) the real center is already trivial. A finite-index subgroup of an ICC group remains ICC: its finite conjugacy class would yield a finite class in the ambient group after using finitely many coset representatives. Thus every group just listed belongs to \(\mathscr K\).

For any such group \(\Gamma\), any countably infinite ICC group \(\Lambda\), and any specified isomorphism \[\theta:L(\Gamma)\longrightarrow pM_k(L(\Lambda))p,\] Theorem 2 yields \[(\mathop{\mathrm{Tr}}_k\otimes\tau)(p)=1,\qquad \theta(u_g)=\chi(g)Wv_{\delta(g)}W^*,\qquad W^*W=1,\quad WW^*=p,\] where \(W\in M_{k,1}(L(\Lambda))\), \(\delta:\Gamma\to\Lambda\) is an actual group isomorphism, and \(\chi:\Gamma\to\mathbb T\) is a character. This includes the specified column in the specified corner. The character becomes the cochain of Equation (1) when scalar twists are present. In particular all the corresponding twisted factors have fundamental group \(\{1\}\).

Definition 1 also includes ICC lattices in the permitted finite-place groups, mixed real and finite-place products, and products with quaternionic or Cayley real factors of property \((T)\). Finite products of the preceding discrete examples give reducible cases: a nonidentity element has a nonidentity coordinate with infinite conjugacy class. More generally the companion exhaustion theorem already allows arbitrary reducible lattices, nonuniform lattices, and torsion. No repetition of the individual geometric proofs is required for these examples.

Recovery from the reduced group C*-algebra

Write \(C_r^*(\Gamma)\) for the operator-norm closure of the group algebra in the left regular representation.

In particular this applies to \(\Gamma=\mathrm{SL}_3(\mathbb Z)\): its reduced group C*-algebra has a unique tracial state by (Bekka 2007, discussion following Corollary 5).

Finite-index endomorphisms of the special linear factor

We state the precise classical fact needed in the final deduction. The finite-index image hypothesis is part of it.