Quasi-BNS invariants

Abstract

We introduce the notion of quasi-BNS invariants, where we replace homomorphisms to $\mathbb{R}$ by homogenous quasimorphisms to $\mathbb{R}$ in the theory of Bieri–Neumann–Strebel invariants. We prove that the quasi-BNS invariant $Q\Sigma (G)$ of a finitely generated group $G$ is open; we connect it to approximate finite generation of almost kernels of homogenous quasimorphisms; finally, we prove a Sikorav-style theorem connecting $Q\Sigma (G)$ to the vanishing of the suitably defined Novikov homology.