The arithmetic category and stable recovery of lattice factors

1. Introduction

A group factor is built from the multiplication of a discrete group, but an isomorphism of factors need not preserve its distinguished group unitaries. We ask when the group, a scalar twist of its multiplication, and the given factor isomorphism can all be recovered. For the lattice class below, we prove this recovery even after amplification and with an arbitrary ICC comparison group.

Connes proved countability of the fundamental group of a property-(T) group factor [9]; his later problems ask whether nonisomorphic ICC property-(T) groups have nonisomorphic factors, and ask for the fundamental groups of those factors [10, Chapter V, Appendix B.\varepsilon, Problems 1–2]. The stronger stable formulation displayed by Popa asks for recovery of the given isomorphism up to a character and inner conjugacy, with amplification scale one [24, Section 3, Equations (3.2) and (3.2')].

The proof uses the more general relation of finite correspondence: two factors act normally and commute on one Hilbert space, with finite module dimensions on both sides. We determine these correspondences through isomorphisms between finite-index subgroups and finite-dimensional projective representations. The representation records the ratio of the two scalar cocycles on the matched subgroups. Neither cocycle separately need have a finite-dimensional realization.

Our algebraic calculation identifies every bounded map, summand, and composition of these arithmetic models for arbitrary countable ICC groups. The geometric exhaustion theorem of the companion [19, Theorem 1.2] makes the models exhaustive for the lattice class. Their integral dimensions recover stable isomorphisms; their compatibility with multiplication also recovers arbitrary finite-index factor neighbors.

1.1. 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 [2].

All scalar cocycles take values in \mathbb T and are normalized:

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

The matrix trace \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\}.

1.2. 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 1.1, 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 [21, Chapter 4, Theorem 1.2]. Independently and concurrently, Zhou constructed another pair [27, Theorem A].

1.3. 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.

1.4. 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

define

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 (1.2) 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

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

For a finite-dimensional \alpha|_{C_d}-projective representation \sigma_d, define

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 (1.2); neither individual endpoint cocycle is assumed finite type.

Theorems 1.2 and 1.3 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.

1.5. 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 [14]. Ioana–Popa–Vaes obtained the first W^*-superrigid groups, in classes of generalized wreath products [15]. Chifan–Ioana–Osin–Sun constructed the first such groups with property (T), using wreath-like products [8]. 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 \Sp(n,1)/\{\pm1\}, n\geq2, Cowling–Haagerup’s completely bounded approximation invariant already determines n from the group factor [11, 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 [1, 22]. Boutonnet–Houdayer developed stationary and noncommutative boundary methods and recovered character rigidity in that framework [6]. 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 [16, 23]. We use Connes’s correspondence category and relative tensor product [10, Chapter V, Appendix B]. Complete finite-index classifications and explicit fusion rules already occur in Vaes’s work on generalized Bernoulli crossed products [26, 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 [12]. 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 [12, 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 [7, 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) [20, Theorem 1.2]. These factors have fundamental group \mathbb R_{>0} [20, Corollary 6.1], in contrast to the trivial fundamental groups in Theorem 1.2.

1.6. 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.

2. Finite modules and the exhaustion interface

Our algebraic calculations need only countability and ICC. The geometric class enters at one precise point: the exhaustion statement below says that the explicit models contain every bifinite correspondence. We first fix the norm convention for their construction.

For completeness, L_\mu(\Gamma) is a factor when \Gamma is ICC: the absolute values of the Fourier coefficients of a central L^2 vector are constant on conjugacy classes, so square summability kills every nonidentity coefficient.

2.1. 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

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 [10, Chapter V, Appendix B, Theorems 13 and 20].

A finite right N-module has a column realization pL^2(N)^n, with dimension (\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 [17, Theorem 10.1.1 and Proposition 10.1.3]. We prove the normality of the graph models directly in Lemmas 3.2 and 3.3. 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.

2.2. The geometric input

The real simple factors in Definition 1.1 have higher rank or are locally \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 [2]. 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 [19, 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.

3. Full-germ correspondence calculus

The calculations in this section are algebraic. We give a group-factor version of the projective induction and cocycle-corrected double-coset calculus in Vaes’s work on generalized Bernoulli crossed products [26, Theorem 2.2 and Section 6.2, especially Definition 6.8 and Theorem 6.9]. We prove the formulas here, including the full-stabilizer argument for every bounded intertwiner and every bimodule summand. Until we apply Proposition 2.1, the groups need only be countable and ICC; the geometric hypotheses of Definition 1.1 are not used. We first construct the arithmetic models with their exact scalar transports, and then determine their intertwiners, summands, conjugates, and relative tensor products. This order also verifies that every model appearing in the exhaustion is a normal bifinite correspondence.

Write

On an M–N correspondence the joint group action is

It has multiplier \alpha. Indeed, its rightmost product is v_{h'}^*v_h^*=(v_hv_{h'})^*, so U_{(g,h)}U_{(g',h')}= \mu(g,g')\overline{\omega(h,h')}U_{(gg',hh')}. In particular the two coordinate actions commute exactly.

3.1. 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

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.

3.2. Projective induction and the arithmetic model

Projective induction, induction in stages, and double-coset tensor decompositions belong to Mackey’s representation-theoretic framework [18, 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

We next identify this construction with the Connes-product model in Theorem 1.4. This also supplies a direct normality check of the subgroup action used to define that product.

For a full stabilizer we write

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

where projective induction uses Equation (3.4) 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

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

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.

3.3. 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 (3.8), combine copies of the same full-germ orbit, and apply Theorem 3.5 to the projection onto K. We obtain a finite orthogonal sum of full-germ bundles, with exactly the intertwiners in Equation (3.11). In particular, if its fibre representations are \rho_d, its actual dimensions are

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.

3.4. Conjugation

On the conjugate correspondence \overline{\mathcal H(C,\gamma)} the endpoint groups are swapped. Its subgroup and base fibre are

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

With swapped coset representatives, the transport is explicitly

Thus it includes the conjugate of the scalar ratio in Equation (3.4). 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\mapsto\overline{\sigma(\delta^{-1}(b))}. Fibre intertwiners are conjugated by the same rule; their adjoints and norms are preserved. Proposition 3.4 also shows directly that the left and right dimensions are exchanged.

3.5. 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

Identify the tensor fibres along this action using

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 (3.16).

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

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

It is the product of the two transports in Equation (3.4). There is no additional factor from the compensating element h, since its lift is the genuine transport in Equation (3.16).

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

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

Hence z=[(x,y)]\mapsto 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 (3.17), with compensating element h=b, gives

The correction has the exact coboundary

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 (3.24) with the multipliers of \sigma and \tau shows that \gamma_m has multiplier \mu|_{A_m}/\theta_m^*\nu. Hence

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 (3.23) exhibits explicitly why replacing the twisted fusion rule by its ordinary version would lose a phase.

The indices in Equation (3.21) satisfy

Counting the left B-orbits in \Lambda/C and the right C-orbits in B\backslash\Lambda gives, respectively,

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.

3.6. 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 3.5 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 (3.16), so descend to the quotient fibres. Under the unitary of Proposition 3.7, 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 (3.14).

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

The counting norms and Equation (3.19) show that it is also the canonical associativity unitary for the relative products. All scalar identifications are those of Equation (3.16); 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

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 3.7 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 2.1 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 1.4, 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.

4. Arbitrary factor endpoints and stable canonical recovery

We now use the correspondence classification to recover the multiplication of an unknown factor. The argument first realizes that factor, up to amplification, as a finite-index extension of the given twisted group factor. Its self-correspondence grading then consists of bounded operators and respects the actual product. This additional multiplication step is what turns the correspondence data into a group presentation. The homogeneous-product and minimal-corner construction has a direct predecessor in Donvil–Vaes [12, Proposition 6.5 and Theorem 6.6]. Their construction already permits arbitrary source cocycles and recovers the exact matched ratio. We adapt it from globally defined automorphism degrees to finite-index isomorphism germs.

4.1. 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=\Comm(\Gamma) for the group of virtual automorphism germs defined in Section 3. The map

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

4.2. 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 4.2 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.

4.3. 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 1.3. 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.

4.4. 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 \Tr_n, with \Tr_n(1)=n.

5. Commensurability, named lattices, and endomorphisms

The category calculation gives concise consequences with different mathematical content. Untwisted correspondence existence detects abstract commensurability. Stable isomorphism recovers an entire group and a specified map. A finite-index endomorphism requires one additional classical covolume argument; its two module dimensions are not initially reciprocal.

5.1. 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 1.3. In particular, its direct proof admits the identity correspondence of L_\mu(\Gamma) whatever the individual cocycle \mu may be.

5.2. Examples and specified canonical maps

The familiar arithmetic examples are \mathrm{SL}_3(\mathbb Z), its principal congruence subgroup

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 [5]; the finite-covolume statement appears also in [4, 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) [2].

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 [3]. Equivalently, the centralizer formulation in [13, 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

Theorem 1.2 yields

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 (1.1) when scalar twists are present. In particular all the corresponding twisted factors have fundamental group \{1\}.

Definition 1.1 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.

5.3. 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 [1, discussion following Corollary 5].

5.4. 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.