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Counterexamples to Baum–Connes and Kadison–Kaplansky

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Disproves the coefficient-free reduced Baum–Connes conjecture through a failure of rational injectivity and a separate failure of surjectivity of assembly. A finitely generated torsion-free example witnesses the injectivity failure. Separately, a torsion-free group has a nontrivial projection in its reduced group C∗-algebra, disproving the Kadison–Kaplansky conjecture.

The release has no Lean statement for this family, so there is no formal statement to compare with the result described above.

The paper states

Theorem 1.1

In An irrational-trace counterexample to reduced Baum–Connes, 23 September 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

Three other possible matches are listed below the card, among them Theorem 1.1 in A torsion-free counterexample to reduced Baum–Connes injectivity and Theorem 1.1 in A torsion-free counterexample to the Kadison–Kaplansky projection conjecture.

There exist a finitely generated discrete group \Gsur and a projection b_0\in C_r^*(\Gsur) with irrational canonical trace. Moreover,

[b_0]\notin\mu_0^{\Gsur}\bigl(K_0^{\Gsur}(\underline E\Gsur)\bigr).

In particular, the coefficient-free reduced Baum–Connes conjecture for countable discrete groups is false.

Three more possible matches

  • Theorem 1.1 in A torsion-free counterexample to reduced Baum–Connes injectivity, 23 September 2026.
  • Theorem 1.1 in A torsion-free counterexample to the Kadison–Kaplansky projection conjecture, 23 September 2026.
  • Theorem 1.2 (Lück) in An irrational-trace counterexample to reduced Baum–Connes, 23 September 2026.
Reported proof verification
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Papers and explanations
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Papers

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An irrational-trace counterexample to reduced Baum–Connes

23 September 2026, current edition.

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We construct a finitely generated discrete group whose reduced group C∗-algebra contains a projection of irrational canonical trace. Its K0-class lies outside the image of the coefficient-free reduced Baum–Connes assembly map. This disproves the coefficient-free reduced Baum–Connes conjecture for countable discrete groups.

Lean statements

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Independent comparisons and proof-verification reports

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  • An unsigned public tier list, the OpenAI Math Tier List, classes it as a counterexample and rates its significance, if it holds up, as “Career-defining publication”. The page says that nothing on it is verified.

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