Mapping tori of free group automorphisms, and the Bieri–Neumann–Strebel invariant of graphs of groups

Cashen, Christopher H.; Levitt, Gilbert

doi:10.1515/jgth-2015-0038

Cited by 16 publications

(15 citation statements)

References 30 publications

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“…Hence we get as special cases P L 2 (G; p 0 ) = P A (G) by Theorem 3.2 (3) and this polytope determines the Alexander norm, and on the other hand P L 2 (G; p ∞ ) = P L 2 (G) which determines the Thurston norm. Theorems 6.2 and 6.3 both rely on the following lemma which follows from the train track theory of Bestvina-Feighn-Handel [BFH]; see [CL,Proposition 5.9] for the argument. Lemma 6.7.…”

Section: Upg Automorphismsmentioning

confidence: 99%

“…By [CL,Remark 5.6], we have b 1 (G) 2 and similarly for G 1 and G 2 . Hence by Theorem 3.2 (2) and (3) as well as the above matrix decomposition, we compute in Wh w (Γ k ) ρ (2) u (G; p k ) = −[p k (I − t · F (g))] + [p k (t − 1)] = −[p k (I − t · F (g 1 ))] − [p k (I − t · F (g 2 ))] + [p k (t − 1)] = (j 1 ) * (ρ (2) u (G 1 ; p 1 k ))) + (j 2 ) * (ρ (2) u (G 2 ; p 2 k ))) − [p k (t − 1)] (6.2)…”

Section: Upg Automorphismsmentioning

confidence: 99%

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Alexander and Thurston norms, and the Bieri–Neumann–Strebel invariants for free-by-cyclic groups

Funke

Kielak

2018

Geom. Topol.

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We investigate Friedl-Lück's universal L 2 -torsion for descending HNN extensions of finitely generated free groups, and so in particular for Fnby-Z groups. This invariant induces a semi-norm on the first cohomology of the group which is an analogue of the Thurston norm for 3-manifold groups.We prove that this Thurston semi-norm is an upper bound for the Alexander semi-norm defined by McMullen, as well as for the higher Alexander seminorms defined by Harvey. The same inequalities are known to hold for 3manifold groups.We also prove that the Newton polytopes of the universal L 2 -torsion of a descending HNN extension of F 2 locally determine the Bieri-Neumann-Strebel invariant of the group. We give an explicit means of computing the BNS invariant for such groups. As a corollary, we prove that the Bieri-Neumann-Strebel invariant of a descending HNN extension of F 2 has finitely many connected components.When the HNN extension is taken over Fn along a polynomially growing automorphism with unipotent image in GL(n, Z), we show that the Newton polytope of the universal L 2 -torsion and the BNS invariant completely determine one another. We also show that in this case the Alexander norm, its higher incarnations, and the Thurston norm all coincide.

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Section: Upg Automorphismsmentioning

confidence: 99%

Section: Upg Automorphismsmentioning

confidence: 99%

Alexander and Thurston norms, and the Bieri–Neumann–Strebel invariants for free-by-cyclic groups

Funke

Kielak

2018

Geom. Topol.

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“…For this, we note that the corresponding tubular group G p,q will have such a finite index subgroup H too, by [6] Proposition 4.3 (iii). Now H will also be a tubular group, but [11] Corollary 2.10 states that a tubular group has a symmetric BNS invariant and so cannot be a strictly ascending HNN extension of any finitely generated group.…”

Section: Corollarymentioning

confidence: 99%

Tubular groups, 1-relator groups and nonpositive curvature

Button

2019

Int. J. Algebra Comput.

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“…In a recent paper [CL,Theorem 5.2] Cashen and Levitt computed Σ(G) for the case where G is the mapping torus of a polynomially growing automorphism φ of F N . They showed that in that situation Σ(G) is centrally symmetric and consists of the complement of finitely many rationally defined hyperplanes in H 1 (G, R).…”

Section: Foliations and Flowsmentioning

confidence: 99%

McMullen polynomials and Lipschitz flows for free-by-cyclic groups

Dowdall

Kapovich²,

Leininger³

2017

J. Eur. Math. Soc.

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Abstract. Consider a group G and an epimorphism u 0 : G → Z inducing a splitting of G as a semidirect product ker(u 0 ) ϕ Z with ker(u 0 ) a finitely generated free group and ϕ ∈ Out(ker(u 0 )) representable by an expanding irreducible train track map. Building on our earlier work [DKL], in which we realized G as π 1 (X) for an Eilenberg-Maclane 2-complex X equipped with a semiflow ψ, and inspired by McMullen's Teichmüller polynomial for fibered hyperbolic 3-manifolds, we construct a polynomial invariant m ∈ Z[H 1 (G; Z)/torsion] for (X, ψ) and investigate its properties.Specifically, m determines a convex polyhedral cone C X ⊂ H 1 (G; R), a convex, real-analytic function H : C X → R, and specializes to give an integral Laurent polynomial mu(ζ) for each integral u ∈ C X . We show that C X is equal to the "cone of sections" of (X, ψ) (the convex hull of all cohomology classes dual to sections of of ψ), and that for each (compatible) cross section Θu ⊂ X with first return map fu : Θu → Θu, the specialization mu(ζ) encodes the characteristic polynomial of the transition matrix of fu. More generally, for every class u ∈ C X there exists a geodesic metric du and a codimension-1 foliation Ωu of X defined by a "closed 1-form" representing u transverse to ψ so that after reparametrizing the flow ψ u s maps leaves of Ωu to leaves via a local e sH(u) -homothety.Among other things, we additionally prove that C X is equal to (the cone over) the component of the BNSinvariant Σ(G) containing u 0 and, consequently, that each primitive integral u ∈ C X induces a splitting of G as an ascending HNN-extension G = Qu * φu with Qu a finite-rank free group and φu : Qu → Qu injective. For any such splitting, we show that the stretch factor of φu is exactly given by e H(u) . In particular, we see that C X and H depend only on the group G and epimorphism u 0 .

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Mapping tori of free group automorphisms, and the Bieri–Neumann–Strebel invariant of graphs of groups

Cited by 16 publications

References 30 publications

Alexander and Thurston norms, and the Bieri–Neumann–Strebel invariants for free-by-cyclic groups

Alexander and Thurston norms, and the Bieri–Neumann–Strebel invariants for free-by-cyclic groups

Tubular groups, 1-relator groups and nonpositive curvature

McMullen polynomials and Lipschitz flows for free-by-cyclic groups

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