Sutured manifolds and $L^2$-Betti numbers

TL;DR

This paper establishes a connection between sutured 3-manifolds being taut and the vanishing of their associated $ ext{L}^2$-Betti numbers, providing new characterizations of Thurston norm minimizing surfaces.

Contribution

It introduces a novel characterization of taut sutured manifolds using $ ext{L}^2$-Betti numbers and applies this to identify Thurston norm minimizing surfaces.

Findings

Taut sutured manifolds correspond to zero $ ext{L}^2$-Betti numbers.

Vanishing $ ext{L}^2$-Betti numbers characterize Thurston norm minimizing surfaces.

The result relies on Agol's virtual fibering theorem.

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.