The group-ring Determinant Conjecture (Luck)

Theorem 1.1: a finitely generated group $G_D$, $n\ge1$ and $A\in M_n(\mathbb Z[G_D])$, invertible over $\mathbb Q[G_D]$ with bounded inverse, whose Fuglede-Kadison determinant lies strictly between $0$ and $1$; the log integral is finite and negative. The group is nonsofic by Elek-Szabo. It disproves the unrestricted conjecture only; it says nothing about torsion-free groups, groups satisfying the Atiyah conjecture, or the determinant approximation conjecture, and it relies on the characteristic-two direct-finiteness counterexample.

Theorem 1.1: a finitely generated group $G_D$, $n\ge1$ and $A\in M_n(\mathbb Z[G_D])$, invertible over $\mathbb Q[G_D]$ with bounded inverse, whose Fuglede-Kadison determinant lies strictly between $0$ and $1$; the log integral is finite and negative. The group is nonsofic by Elek-Szabo. It disproves the unrestricted conjecture only; it says nothing about torsion-free groups, groups satisfying the Atiyah conjecture, or the determinant approximation conjecture, and it relies on the characteristic-two direct-finiteness counterexample.