Vanishing of the $L^2$-torsion polytope for amenable groups

Let $G\neq \mathbb{Z}$ be an amenable group satisfying the Atiyah Conjecture. Suppose that $G$ is of type $F$ and that $\operatorname{Wh}(G)=0$. The $L^2$-torsion polytope $P(G)$ is defined under these hypotheses and takes values in the integral polytope group $\mathcal{P}_T(H_1(G)_f)$. Vanishing conjecture.

$$P(G)=0.$$

This conjecture predicts the vanishing of the $L^2$-torsion polytope for amenable groups under the stated finiteness, Atiyah-Conjecture, and Whitehead-group assumptions. The paper proves this in particular for elementary amenable groups of type $F$, while the general amenable case remains open.