Sutured manifolds and $L^2$-Betti numbers

Abstract

Using the virtual fibering theorem of Agol we show that a sutured 3-manifold $(M, R_+,R_-,\gamma)$ is taut if and only if the $\ell^2$-Betti numbers of the pair $(M,R_-)$ are zero. As an application we can characterize Thurston norm minimizing surfaces in a 3-manifold $N$ with empty or toroidal boundary by the vanishing of certain $\ell^2$-Betti numbers.