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The group-ring Determinant Conjecture

Must the Fuglede–Kadison determinant of an integral group-ring matrix satisfy the conjectured lower bound of one? A decisive target is a finitely generated group G and A∈Z[G] invertible over Q[G] whose determinant lies strictly between zero and one.

Why it matters and what counts as progress

Why it matters: The determinant question extends the lower bound for nonsingular integer matrices to integral matrices over arbitrary group rings.

What counts as progress: Construct and evaluate such a determinant witness or prove the lower bound for additional group classes.

How the problem developed

  1. 2002 · Conjecture context [1]

    Wolfgang Lück Formulated the determinant lower-bound problem for integral group-ring matrices, extending the integer-matrix bound to operators on group Hilbert spaces.

  2. 23 Sept 2026 · Counterexample [2]

    OpenAI Claimed a finitely generated group and integral group-ring matrix invertible over the rational group ring whose Fuglede–Kadison determinant lies strictly between zero and one.

AI contribution

Sources

  1. Wolfgang Lück, L²-Invariants: Theory and Applications to Geometry and K-Theory
  2. OpenAI, A Counterexample to the Group-Ring Determinant Conjecture
  3. A Counterexample to the Group-Ring Determinant Conjecture
  4. Group-ring determinant counterexample manuscript
  5. Lean determinant theorem
  6. Formalization scope
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