The Bieri-Neumann-Strebel invariants via Newton polytopes

TL;DR

This paper explores the relationship between Newton polytopes of determinants over twisted Laurent polynomial rings and Bieri-Neumann-Strebel invariants, providing new proofs, extensions, and applications in geometric group theory and topology.

Contribution

It establishes that Newton polytopes of such determinants are single polytopes, links them to BNS invariants, and applies these results to 3-manifold groups, free-by-cyclic groups, and Poincaré duality groups, confirming conjectures.

Findings

Newton polytopes of determinants are single polytopes.

Reproved Thurston's theorem on BNS invariants of 3-manifold groups.

Extended results to free-by-cyclic and descending HNN extension groups.

Abstract

We study the Newton polytopes of determinants of square matrices defined over rings of twisted Laurent polynomials. We prove that such Newton polytopes are single polytopes (rather than formal differences of two polytopes); this result can be seen as analogous to the fact that determinants of matrices over commutative Laurent polynomial rings are themselves polynomials, rather than rational functions. We also exhibit a relationship between the Newton polytopes and invertibility of the matrices over Novikov rings, thus establishing a connection with the invariants of Bieri-Neumann-Strebel (BNS) via a theorem of Sikorav. We offer several applications: we reprove Thurston's theorem on the existence of a polytope controlling the BNS invariants of a $3$-manifold group; we extend this result to free-by-cyclic groups, and the more general descending HNN extensions of free groups. We also show that the BNS invariants of Poincar\'e duality groups of type $\mathtt{F}$ in dimension $3$ and groups of deficiency one are determined by a polytope, when the groups are assumed to be agrarian, that is their integral group rings embed in skew-fields. The latter result partially confirms a conjecture of Friedl. We also deduce the vanishing of the Newton polytopes associated to elements of the Whitehead groups of many groups satisfying the Atiyah conjecture. We use this to show that the $L^2$-torsion polytope of Friedl-L\"uck is invariant under homotopy. We prove the vanishing of this polytope in the presence of amenability, thus proving a conjecture of Friedl-L\"uck-Tillmann.