A Counterexample to the Group-Ring Determinant Conjecture

We disprove the unrestricted group-ring Determinant Conjecture. We construct a finitely generated group G and a square matrix over $\mathbb{Z}[G]$ that is invertible over $\mathbb{Q}[G]$ and has Fuglede–Kadison determinant strictly between zero and one. The logarithmic integral defining the determinant is finite.