Thurston norm, polytopes and splitting complexity

We show that if $G$ is a finitely generated torsion-free group satisfying the Strong Atiyah Conjecture with vanishing first $L^{2}$-Betti number, then the map that assigns to each surjective integral character the first $L^2$-Betti number of the kernel extends to a seminorm on the first cohomology group of $G$ with real coefficients. We call this seminorm the Thurston norm. Moreover, we show that this norm is induced by a polytope in the first homology group with real coefficients. We also generalize this result to higher $L^{2}$-Betti numbers of the kernels, thereby confirming a conjecture of Friedl, L\"uck and Tillmann. In the case where $G$ is either a free-by-cyclic group or the fundamental group of an admissible $3$-manifold, we show that the Thurston norm of $G$ admits a combinatorial interpretation that relates it to the splitting complexity of the character. This confirms a conjecture of Gardam and Kielak. As an application, we show that there exists an algorithm to compute the Bieri--Neumann--Strebel invariant of free-by-cyclic groups, and discuss connections to the isomorphism problem in free-by-cyclic groups.