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Free-by-cyclic group

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In group theory, especially, in geometric group theory, the class of free-by-cyclic groups have been deeply studied as important examples. A group G {\displaystyle G} is said to be free-by-cyclic if it has a free normal subgroup F {\displaystyle F} such that the quotient group G / F {\displaystyle G/F} is cyclic. In other words, G {\displaystyle G} is free-by-cyclic if it can be expressed as a group extension of a free group by a cyclic group (NB there are two conventions for 'by'). Usually, we assume F {\displaystyle F} is finitely generated and the quotient is an infinite cyclic group. Equivalently, we can define a free-by-cyclic group constructively: if φ {\displaystyle \varphi } is an automorphism of F {\displaystyle F} , the semidirect product F ⋊ φ Z {\displaystyle F\rtimes _{\varphi }\mathbb {Z} } is a free-by-cyclic group.

An isomorphism class of a free-by-cyclic group is determined by an outer automorphism. If two automorphisms φ , ψ {\displaystyle \varphi ,\psi } represent the same outer automorphism, that is, φ = ψ ι {\displaystyle \varphi =\psi \iota } for some inner automorphism ι {\displaystyle \iota } , the free-by-cyclic groups F ⋊ φ Z {\displaystyle F\rtimes _{\varphi }\mathbb {Z} } and F ⋊ ψ Z {\displaystyle F\rtimes _{\psi }\mathbb {Z} } are isomorphic.

Examples and results

The study of free-by-cyclic groups is strongly related to that of the attaching outer automorphism. Among the motivating questions are those concerning their non-positive curvature properties, such as being CAT(0).

  • A free-by-cyclic group is hyperbolic, if and only if it does not contain a subgroup isomorphic to Z 2 {\displaystyle \mathbb {Z} ^{2}} , if and only if no nontrivial conjugacy class is left invariant by the attaching automorphism (irreducible case: Bestvina and Feighn, 1992; general case: Brinkmann, 2000).
  • Hyperbolic free-by-cyclic groups are fundamental groups of compact non-positively curved cube complexes (Hagen and Wise, 2015).
  • Some free-by-cyclic groups are hyperbolic relative to free-abelian subgroups. More generally, all free-by-cyclic groups are hyperbolic relative to a collection of subgroups that are free-by-cyclic for an automorphism of polynomial growth.
  • Any finitely generated subgroup of a free-by-cyclic group is finitely presented (Feighn and Handel, 1999).
  • The conjugacy problem for free-by-cyclic groups is solved (Bogopolski, Martino, Maslakova and Ventura, 2006).
  • Notably, there are non-CAT(0) free-by-cyclic groups (Gersten, 1994).
  • However, all free-by-cyclic groups satisfy a quadratic isoperimetric inequality (Bridson and Groves, 2010).
  • All free-by-cyclic groups where the underlying free group has rank 2 {\displaystyle 2} are CAT(0) (Brady, 1995).
  • Many examples of free-by-cyclic groups with polynomially-growing attaching maps are known to be CAT(0).
  • Free-by-cyclic groups are equationally noetherian and have well-ordered growth rates (Kudlinska, Valiunas, 2024 preprint).

References

  1. Brinkmann, P. (2000-12-01). "Hyperbolic automorphisms of free groups". Geometric and Functional Analysis. 10 (5): 1071–1089. doi:10.1007/PL00001647. ISSN 1420-8970.
  2. Hagen, Mark F.; Wise, Daniel T. (2015-02-01). "Cubulating hyperbolic free-by-cyclic groups: the general case". Geometric and Functional Analysis. 25 (1): 134–179. doi:10.1007/s00039-015-0314-y. ISSN 1420-8970.
  3. Ghosh, Pritam (2023). "Relative hyperbolicity of free-by-cyclic extensions". Compositio Mathematica. 159 (1): 153–183. arXiv:1802.08570. doi:10.1112/S0010437X22007813. ISSN 0010-437X.
  4. Dahmani, François; Li, Ruoyu (2022). "Relative hyperbolicity for automorphisms of free products and free groups". Journal of Topology and Analysis. 14 (1): 55–92. arXiv:1901.06760. doi:10.1142/S1793525321500011. ISSN 1793-5253.
  5. Feighn, Mark; Handel, Michael (1999). "Mapping Tori of Free Group Automorphisms are Coherent". Annals of Mathematics. 149 (3): 1061–1077. arXiv:math/9905209. doi:10.2307/121081. ISSN 0003-486X. JSTOR 121081.
  6. Bogopolski, O.; Martino, A.; Maslakova, O.; Ventura, E. (2006). "The conjugacy problem is solvable in free-by-cyclic groups". Bulletin of the London Mathematical Society. 38 (5): 787–794. doi:10.1112/S0024609306018674. ISSN 0024-6093.
  7. Gersten, S. M. (1994). "The automorphism group of a free group is not a CAT(0) group". Proceedings of the American Mathematical Society. 121 (4): 999–1002. doi:10.2307/2161207. ISSN 0002-9939. JSTOR 2161207.
  8. Bridson, Martin; Groves, Daniel (2010). "The quadratic isoperimetric inequality for mapping tori of free group automorphisms". American Mathematical Society. Retrieved 2024-11-02.
  9. Brady, Thomas (1995-05-26). "Complexes of nonpositive curvature for extensions of F2 by Z". Topology and Its Applications. 63 (3): 267–275. doi:10.1016/0166-8641(94)00072-B. ISSN 0166-8641.
  10. Samuelson, Peter (2006-09-01). "On CAT(0) structures for free-by-cyclic groups". Topology and Its Applications. 153 (15): 2823–2833. doi:10.1016/j.topol.2005.12.002. ISSN 0166-8641.
  11. Lyman, Rylee Alanza (2023). "Some New CAT(0) Free-by-Cyclic Groups". Michigan Mathematical Journal. 73 (3): 621–630. arXiv:1909.03097. doi:10.1307/mmj/20205989. ISSN 0026-2285.
  12. "Free-by-cyclic groups are equationally Noetherian". arxiv.org. Retrieved 2024-11-02.
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