The conjecture on $L^2$-torsion for aspherical manifolds

Let $M$ be a closed aspherical manifold of odd dimension and let $\widetilde{M}$ be its universal covering. $L^2$-torsion conjecture. The space $\widetilde{M}$ is determinant-$L^2$-acyclic and $(-1)^{(\dim(M)-1)/2}\rho^{(2)}(\widetilde{M})\geq0$. For a closed connected odd-dimensional Riemannian manifold with negative sectional curvature, the same determinant-$L^2$-acyclicity holds and the inequality is strict. If an aspherical closed manifold has fundamental group containing an amenable infinite normal subgroup, then $\widetilde{M}$ is determinant-$L^2$-acyclic and $\rho^{(2)}(\widetilde{M})=0$. These assertions propose sign and vanishing laws for $L^2$-torsion in geometric settings.