Quasi-BNS invariants

TL;DR

This paper introduces quasi-BNS invariants by replacing homomorphisms with homogeneous quasimorphisms, establishing their topological properties, connections to approximate kernels, and relations to Novikov homology.

Contribution

It defines quasi-BNS invariants, proves their openness, links them to approximate kernel finite generation, and relates them to Novikov homology.

Findings

Quasi-BNS invariant $Q\Sigma(G)$ is open for finitely generated groups.

Connections established between $Q\Sigma(G)$ and approximate finite generation.

A Sikorav-style theorem relates $Q\Sigma(G)$ to Novikov homology.

Abstract

We introduce the notion of quasi-BNS invariants, where we replace homomorphism 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.