Incoherence and fibering of many free-by-free groups
Annales de l'Institut Fourier, Online first, 13 p.

We show that free-by-free groups satisfying a homological criterion, which we call excessive homology, are incoherent. This class is large in nature, including many examples of hyperbolic and non-hyperbolic free-by-free groups. We apply this criterion to finite index subgroups of F 2 F n to show incoherence of all such groups, and to other similar classes of groups. Furthermore, we show that a large class of groups, including free-by-free, surface-by-surface, and finitely generated-by-RAAG, algebraically fiber if they have excessive homology.

Nous montrons que les extensions d’un groupe libre par un groupe libre satisfaisant un critère homologique, que nous appelons “ homologie excessive ”, sont incohérents. Cette classe est large et comprend de nombreux exemples de groupes hyperboliques et non hyperboliques. Nous appliquons ce critère aux sous-groupes d’indice fini de F 2 F n pour montrer l’incohérence de tous ces groupes, et à d’autres classes de groupes similaires. De plus, nous montrons que les groupes d’une plus vaste classe, incluant les extensions d’un groupe libre par un groupe libre, d’un groupe de surface par un groupe de surface, d’un groupe de type fini par un groupe d’Artin á angles droits, fibrent algébriquement s’ils ont une homologie excessive.

Online First:
DOI: 10.5802/aif.3494
Classification: 20F65
Keywords: incoherence, free-by-free, surface-by-free, algebraic fibering
Kropholler, Robert P. 1; Walsh, Genevieve S. 2

1 Mathematics Institute Zeeman Building University of Warwick Coventry, England
2 Department of Mathematics Tufts University Medford, MA USA
     author = {Kropholler, Robert P. and Walsh, Genevieve S.},
     title = {Incoherence and fibering of many free-by-free groups},
     journal = {Annales de l'Institut Fourier},
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     year = {2022},
     doi = {10.5802/aif.3494},
     language = {en},
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Kropholler, Robert P.; Walsh, Genevieve S. Incoherence and fibering of many free-by-free groups. Annales de l'Institut Fourier, Online first, 13 p.

[1] Agol, Ian The virtual Haken conjecture, Doc. Math., Volume 18 (2013), pp. 1045-1087 (With an appendix by Agol, Daniel Groves, and Jason Manning) | DOI | MR | Zbl

[2] Bieri, Robert Normal subgroups in duality groups and in groups of cohomological dimension 2, J. Pure Appl. Algebra, Volume 7 (1976) no. 1, pp. 35-51 | DOI | MR | Zbl

[3] Bieri, Robert; Neumann, Walter D.; Strebel, Ralph A geometric invariant of discrete groups, Invent. Math., Volume 90 (1987) no. 3, pp. 451-477 | DOI | MR | Zbl

[4] Bowditch, Brian H.; Mess, Geoffrey A 4-dimensional Kleinian group, Trans. Amer. Math. Soc., Volume 344 (1994) no. 1, pp. 391-405 | DOI | MR | Zbl

[5] Bridson, Martin R.; Haefliger, André Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 319, Springer-Verlag, Berlin, 1999, xxii+643 pages | DOI | MR | Zbl

[6] Davis, Michael W.; Okun, Boris Vanishing theorems and conjectures for the 2 -homology of right-angled Coxeter groups, Geom. Topol., Volume 5 (2001), pp. 7-74 | DOI | MR | Zbl

[7] Feighn, Mark; Handel, Michael Mapping tori of free group automorphisms are coherent, Ann. of Math. (2), Volume 149 (1999) no. 3, pp. 1061-1077 | DOI | MR | Zbl

[8] Gersten, Stephen M. The automorphism group of a free group is not a CAT (0) group, Proc. Amer. Math. Soc., Volume 121 (1994) no. 4, pp. 999-1002 | DOI | MR | Zbl

[9] Gordon, Cameron McA Artin groups, 3-manifolds and coherence, Bol. Soc. Mat. Mexicana (3), Volume 10 (2004) no. Special Issue, pp. 193-198 | MR | Zbl

[10] Haglund, Frédéric Finite index subgroups of graph products, Geom. Dedicata, Volume 135 (2008), pp. 167-209 | DOI | MR | Zbl

[11] Haglund, Frédéric; Wise, Daniel T. Special cube complexes, Geom. Funct. Anal., Volume 17 (2008) no. 5, pp. 1551-1620 | DOI | MR | Zbl

[12] Karrass, Abraham; Solitar, Donald The subgroups of a free product of two groups with an amalgamated subgroup, Trans. Amer. Math. Soc., Volume 150 (1970), pp. 227-255 | DOI | MR | Zbl

[13] Rips, Eliyahu Subgroups of small cancellation groups, Bull. London Math. Soc., Volume 14 (1982) no. 1, pp. 45-47 | DOI | MR | Zbl

[14] Scott, G. Peter Finitely generated 3-manifold groups are finitely presented, J. London Math. Soc. (2), Volume 6 (1973), pp. 437-440 | DOI | MR | Zbl

[15] Strebel, Ralph Notes on the Sigma invariants (2013) (

[16] Swenson, Eric L. Quasi-convex groups of isometries of negatively curved spaces, Topology Appl., Volume 110 (2001) no. 1, pp. 119-129 Geometric topology and geometric group theory (Milwaukee, WI, 1997) | DOI | MR | Zbl

[17] Thurston, William P. A norm for the homology of 3-manifolds, Mem. Amer. Math. Soc., Volume 59 (1986) no. 339, p. i-vi and 99–130 | MR | Zbl

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