Connes' rigidity conjecture [connes-000O]

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For a countable discrete group \(\Gamma \), the group von Neumann algebra \(L(\Gamma )\) is generated by the left regular representation of \(\Gamma \). When \(\Gamma \) has infinite conjugacy classes (ICC), meaning that every nonidentity conjugacy class is infinite, \(L(\Gamma )\) is a \(\mathrm {II}_1\) factor. The reconstruction question asks how much of \(\Gamma \) remains visible in that analytic completion.

Kazhdan's property (T) says that a unitary representation with almost invariant unit vectors has a nonzero invariant vector [ershov2010noncommutative, Section 3.1]. ICC makes the group von Neumann algebra a factor, while property (T) imposes representation-theoretic rigidity. These are the two hypotheses in the conjecture and in all three counterexample constructions below [zhou2026icc, Section 1, pp. 1–2].

Connes first proved that, for an ICC property-(T) group, \(\operatorname {Out}(L(\Gamma ))\) and the fundamental group \(\mathcal F(L(\Gamma ))\) are countable [zhou2026icc, Section 1, p. 1].

He subsequently proposed that such groups should be \(W^*\)-superrigid: for every countable group \(\Lambda \), an isomorphism \(L(\Lambda )\cong L(\Gamma )\) should force \(\Lambda \cong \Gamma \). Zhou's introduction states this original 1982 conjecture and distinguishes it from the later, narrower rigidity question for higher-rank lattices [zhou2026icc, Section 1, pp. 1–2].

The counterexamples considered here disprove the broad conjecture. They do not settle that higher-rank-lattice question.