Geometry & Topology

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

Florian Funke and Dawid Kielak

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We investigate Friedl and Lück’s universal L2–torsion for descending HNN extensions of finitely generated free groups, and so in particular for Fn-by- groups. This invariant induces a seminorm on the first cohomology of the group which is an analogue of the Thurston norm for 3–manifold groups.

We prove that this Thurston seminorm is an upper bound for the Alexander seminorm defined by McMullen, as well as for the higher Alexander seminorms defined by Harvey. The same inequalities are known to hold for 3–manifold groups.

We also prove that the Newton polytopes of the universal L2–torsion of a descending HNN extension of F2 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 F2 has finitely many connected components.

When the HNN extension is taken over Fn along a polynomially growing automorphism with unipotent image in GL(n,), we show that the Newton polytope of the universal L2–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.

Article information

Geom. Topol., Volume 22, Number 5 (2018), 2647-2696.

Received: 31 May 2016
Revised: 26 October 2017
Accepted: 14 January 2018
First available in Project Euclid: 26 March 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]
Secondary: 16S85: Rings of fractions and localizations [See also 13B30] 20E06: Free products, free products with amalgamation, Higman-Neumann- Neumann extensions, and generalizations

free-by-cyclic groups ascending HNN extensions of free groups BNS invariants Thurston norm Alexander norm


Funke, Florian; Kielak, Dawid. Alexander and Thurston norms, and the Bieri–Neumann–Strebel invariants for free-by-cyclic groups. Geom. Topol. 22 (2018), no. 5, 2647--2696. doi:10.2140/gt.2018.22.2647.

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  • J W Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc. 30 (1928) 275–306
  • L Bartholdi, Amenability of groups is characterized by Myhill's theorem, with an appendix by D Kielak, preprint (2016) To appear in J. Eur. Math. Soc.
  • M Bestvina, M Feighn, M Handel, The Tits alternative for ${\rm Out}(F_n)$, I: Dynamics of exponentially-growing automorphisms, Ann. of Math. 151 (2000) 517–623
  • R Bieri, W D Neumann, R Strebel, A geometric invariant of discrete groups, Invent. Math. 90 (1987) 451–477
  • C H Cashen, G Levitt, Mapping tori of free group automorphisms, and the Bieri–Neumann–Strebel invariant of graphs of groups, J. Group Theory 19 (2016) 191–216
  • J C Cha, S Friedl, F Funke, The Grothendieck group of polytopes and norms, Münster J. Math. 10 (2017) 75–81
  • M Clay, $\ell^2$–torsion of free-by-cyclic groups, Q. J. Math. 68 (2017) 617–634
  • P M Cohn, Free ideal rings and localization in general rings, New Mathematical Monographs 3, Cambridge Univ. Press (2006)
  • W Dicks, P Menal, The group rings that are semifirs, J. London Math. Soc. 19 (1979) 288–290
  • J Dieudonné, Les déterminants sur un corps non commutatif, Bull. Soc. Math. France 71 (1943) 27–45
  • J Dubois, S Friedl, W Lück, The $L^2$–Alexander torsion of $3$–manifolds, preprint (2014)
  • N M Dunfield, Alexander and Thurston norms of fibered $3$–manifolds, Pacific J. Math. 200 (2001) 43–58
  • R H Fox, Free differential calculus, I: Derivation in the free group ring, Ann. of Math. 57 (1953) 547–560
  • S Friedl, W Lück, The $L^2$–torsion function and the Thurston norm of $3$–manifolds, preprint (2015)
  • S Friedl, W Lück, $L^2$–Euler characteristics and the Thurston norm, preprint (2016)
  • S Friedl, W Lück, Universal $L^2$–torsion, polytopes and applications to $3$–manifolds, Proc. Lond. Math. Soc. 114 (2017) 1114–1151
  • S Friedl, S Tillmann, Two-generator one-relator groups and marked polytopes, preprint (2015)
  • F Funke, The integral polytope group, preprint (2016)
  • F Funke, The $L^2$–torsion polytope of amenable groups, preprint (2017)
  • R Geoghegan, M L Mihalik, M Sapir, D T Wise, Ascending HNN extensions of finitely generated free groups are Hopfian, Bull. London Math. Soc. 33 (2001) 292–298
  • M Hall, Jr, Subgroups of finite index in free groups, Canadian J. Math. 1 (1949) 187–190
  • S L Harvey, Higher-order polynomial invariants of $3$–manifolds giving lower bounds for the Thurston norm, Topology 44 (2005) 895–945
  • S L Harvey, Monotonicity of degrees of generalized Alexander polynomials of groups and $3$–manifolds, Math. Proc. Cambridge Philos. Soc. 140 (2006) 431–450
  • S Harvey, S Friedl, Non-commutative multivariable Reidemester torsion and the Thurston norm, Algebr. Geom. Topol. 7 (2007) 755–777
  • G Higman, The units of group-rings, Proc. London Math. Soc. 46 (1940) 231–248
  • D H Kochloukova, Some Novikov rings that are von Neumann finite and knot-like groups, Comment. Math. Helv. 81 (2006) 931–943
  • P Linnell, W Lück, Localization, Whitehead groups, and the Atiyah conjecture, preprint (2016)
  • W Lück, $L^2$–invariants: theory and applications to geometry and $K$–theory, Ergeb. Math. Grenzgeb. 44, Springer (2002)
  • W Lück, Twisting $L^2$–invariants with finite-dimensional representations, preprint (2015)
  • W Lück, T Schick, $L^2$–torsion of hyperbolic manifolds of finite volume, Geom. Funct. Anal. 9 (1999) 518–567
  • A I Mal'cev, On the embedding of group algebras in division algebras, Doklady Akad. Nauk SSSR 60 (1948) 1499–1501 In Russian
  • C T McMullen, The Alexander polynomial of a $3$–manifold and the Thurston norm on cohomology, Ann. Sci. École Norm. Sup. 35 (2002) 153–171
  • B H Neumann, On ordered division rings, Trans. Amer. Math. Soc. 66 (1949) 202–252
  • J-C Sikorav, Homologie de Novikov associée à une classe de cohomologie réelle de degré un, Thèse d'Etat, Universit'e Paris-Sud XI, Orsay (1987) Available at \setbox0\makeatletter\@url {\unhbox0
  • J R Silvester, Introduction to algebraic $K$–theory, Chapman & Hall, London (1981)
  • J R Stallings, Topology of finite graphs, Invent. Math. 71 (1983) 551–565
  • R Strebel, Notes on the Sigma invariants, preprint (2012)
  • D Tamari, A refined classification of semi-groups leading to generalized polynomial rings with a generalized degree concept, from “Proceedings of the International Congress of Mathematicians” (J C H Gerretsen, J de Groot, editors), volume I, North-Holland, Amsterdam (1957) 439–440
  • W P Thurston, A norm for the homology of $3$–manifolds, Mem. Amer. Math. Soc. 339, Amer. Math. Soc., Providence, RI (1986) 99–130
  • F Waldhausen, Algebraic $K$–theory of generalized free products, III, IV, Ann. of Math. 108 (1978) 205–256
  • C Wegner, The Farrell–Jones conjecture for virtually solvable groups, J. Topol. 8 (2015) 975–1016