A counterexample to the determinant approximation conjecture

Abstract

The determinant approximation conjecture states that for a residually finite group $G$ and a chain $G_i$ of finite index normal subgroups with trivial intersection, the Fuglede-Kadison determinant of every matrix over the group algebra $\mathbb{Q}G$ is the limit of the Fuglede-Kadison determinants of the matrix reductions from $\mathbb{Q}G$ to $\mathbb{Q}(G/G_i)$. We disprove the conjecture by exhibiting an explicit element of the integral group ring of the Heisenberg group with determinant equal to $2$ and reduced determinants bounded by $\sqrt[3]{5}$.

arXiv comment16 pages