Thurston norm, polytopes and splitting complexity

Abstract

Let $G$ be a finitely generated torsion-free group satisfying the Strong Atiyah Conjecture. Assuming that $b_1^{(2)}(G)=0$, we associate to each surjective integral character $\phi$ the quantity $b_1^{(2)}(\ker \phi)$. We prove that the resulting function extends to a seminorm on $H^1(G;\mathbb{R})$ and is the thickness function of a polytope in $H_1(G;\mathbb{R})$. More generally, assuming that $G$ is of type $\mathrm{FP}_k(\mathbb{Q})$ and has $b_k^{(2)}(G)=0$, we prove that the function assigning $b_k^{(2)}(\ker \phi)$ to each surjective integral character $\phi$ is likewise the thickness function of a polytope. This confirms a conjecture of Friedl, L\"uck, and Tillmann. For torsion-free virtually (finitely generated free)-by-cyclic groups, we relate the resulting invariant to the $L^2$-Betti complexity of a character, thereby confirming a conjecture of Gardam and Kielak. As an application, we establish the existence of an algorithm for computing the Bieri--Neumann--Strebel invariant of free-by-cyclic groups and discuss connections with the isomorphism problem for such groups.