The Bieri–Neumann–Strebel invariants via Newton polytopes

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 Poincaré 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–Lück is invariant under homotopy. We prove the vanishing of this polytope in the presence of amenability, thus proving a conjecture of Friedl–Lück–Tillmann.

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Acknowledgements

The author would like to thank Kai-Uwe Bux for comments on an earlier version of the article, as well as Stefan Friedl, Fabian Henneke, Wolfgang Lück, and Stefan Witzel for helpful discussions. The author was supported by the Priority Programme 2026 ‘Geometry at infinity’ of the German Science Foundation (DFG).

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Kielak, D. The Bieri–Neumann–Strebel invariants via Newton polytopes. Invent. math. 219, 1009–1068 (2020). https://doi.org/10.1007/s00222-019-00919-9

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