Characteristic-module obstruction [connes-001K]
✍️source

    Theorem. Characteristic modules detect nonisomorphism [connes-0008]
    firstProduct_semisimplepaperCharacteristicModuleEquivpaperGroups_not_isomorphic
    L∃∀N

      The Section 6 invariant is semisimplicity of a module obtained from the characteristic abelian kernel. This is suitable for formal transport: a group isomorphism identifies the characteristic kernels, descends to an automorphism \(\sigma \) of the finite quotient, and induces a linear equivalence between the first quotient module and the second module with its action restricted along \(\sigma \). Semisimplicity is invariant under that linear equivalence.

      The characteristic-kernel and quotient-twist boundaries used here are separated in § [connes-000H].

      On the first side, explicit decomposition into simple summands culminates in Connes.PaperModuleSemisimple.firstProduct_semisimple.

      On the second side, a nonsplit submodule obstruction is proved for every quotient twist \(\sigma \). The bridge Connes.PaperNonisomorphism.paperCharacteristicModuleEquiv turns a hypothetical group isomorphism into the forbidden module equivalence.

      The public contradiction is Connes.PaperModuleSemisimpleTransport.paperGroups_not_isomorphic. This records the invariant used in [zhou2026icc, Section 6] while keeping the group-theoretic, finite-certificate, and module-theoretic arguments in separate files.

      References

        Reference. ICC property-(T) groups without W-star-superrigidity [zhou2026icc]

        @misc{zhou2026icc,
  title = {ICC Property-(T) Groups Without W*-Superrigidity},
  author = {Zhou, Shuoxing},
  year = {2026},
  eprint = {2608.02327},
  archivePrefix = {arXiv},
  primaryClass = {math.OA},
  url = {https://arxiv.org/abs/2608.02327}
}

        Context

          Nonisomorphism by characteristic modules (Zhou §6) [connes-0015]

            Characteristic-module obstruction [connes-001K]
            ✍️source

              Theorem. Characteristic modules detect nonisomorphism [connes-0008]
              firstProduct_semisimplepaperCharacteristicModuleEquivpaperGroups_not_isomorphic
              L∃∀N

                The Section 6 invariant is semisimplicity of a module obtained from the characteristic abelian kernel. This is suitable for formal transport: a group isomorphism identifies the characteristic kernels, descends to an automorphism \(\sigma \) of the finite quotient, and induces a linear equivalence between the first quotient module and the second module with its action restricted along \(\sigma \). Semisimplicity is invariant under that linear equivalence.

                The characteristic-kernel and quotient-twist boundaries used here are separated in § [connes-000H].

                On the first side, explicit decomposition into simple summands culminates in Connes.PaperModuleSemisimple.firstProduct_semisimple.

                On the second side, a nonsplit submodule obstruction is proved for every quotient twist \(\sigma \). The bridge Connes.PaperNonisomorphism.paperCharacteristicModuleEquiv turns a hypothetical group isomorphism into the forbidden module equivalence.

                The public contradiction is Connes.PaperModuleSemisimpleTransport.paperGroups_not_isomorphic. This records the invariant used in [zhou2026icc, Section 6] while keeping the group-theoretic, finite-certificate, and module-theoretic arguments in separate files.

                Characteristic kernels and quotient-twisted transport [connes-000H]

                  The characteristic-kernel step maps an abelian normal subgroup to both factors of \(\mathrm {SL}_3(\mathbb F_2[t])\times \operatorname {Sp}_4(\mathbb F_2)\). It therefore needs two vanishing results:

                  The product argument applying both inputs is paperSemidirectNormalAbelian_le_kernel. This confines finite enumeration to the symplectic factor and keeps both factor-specific proofs out of the abstract module transport.

                  Design adaptation: twisting does not repair nonsplitting. Restriction along a group automorphism is an equivalence of module categories, with inverse given by restriction along the inverse automorphism. The formal proof nevertheless exports the exact twisted obstruction needed by the consumer, so no unproved categorical transport principle remains hidden at the final contradiction.