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 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.

Residually finite rationally solvable groups and virtual fibring

“…We will now discuss the behaviour of D(G) under passing to subgroups of G. Sketch proof. Since the mathematical content of this proposition is standard (see for example [Kie,Proposition 4.6] or [Lüc02, Section 10.2]), we offer only a sketch proof.…”

The agrarian polytope of two‐generator one‐relator groups

“…It is easily verified in [17, Lemma 3.12] (and the discussion following the lemma) that is a well‐defined group homomorphisms.…”

“…The second author introduced the notion of an agrarian group in [17]. In [14], the authors then developed a theory of algebraic invariants of nice spaces with an action of an agrarian group, which proceeds in analogy to the construction of ‐invariants.…”

“…Examples of agrarian groups are given by torsion‐free groups satisfying the Atiyah conjecture over [20, Theorem 10.39] as well as by torsion‐free one‐relator groups [18]. For a more detailed discussion of examples and the inheritance properties enjoyed by agrarian groups, see [17].…”

On the Bieri–Neumann–Strebel–Renz invariants of residually free groups

“…In general it is not easy to calculate the Σ-invariants, but in some cases their description or some information about them is known: these cases include right-angled Artin groups (RAAGs) [31], some Artin groups that are not RAAGs [2,3], the Thompson group F [15], generalized Thompson groups F n,∞ [27,43], free-by-cyclic groups [20], Poincare duality groups of dimension 3 [25] and limit groups [26].…”