Free-by-cyclic group

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

An isomorphism class of a free-by-cyclic group is determined by an outer automorphism. If two automorphisms ${\displaystyle \varphi ,\psi }$ represent the same outer automorphism, that is, ${\displaystyle \varphi =\psi \iota }$ for some inner automorphism ${\displaystyle \iota }$, the free-by-cyclic groups ${\displaystyle F\rtimes _{\varphi }\mathbb {Z} }$ and ${\displaystyle F\rtimes _{\psi }\mathbb {Z} }$ 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 ${\displaystyle \mathbb {Z} ^{2}}$, 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). ^[1]
• Hyperbolic free-by-cyclic groups are fundamental groups of compact non-positively curved cube complexes (Hagen and Wise, 2015). ^[2]
• 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. ^{[3] [4]}
• Any finitely generated subgroup of a free-by-cyclic group is finitely presented (Feighn and Handel, 1999). ^[5]
• The conjugacy problem for free-by-cyclic groups is solved (Bogopolski, Martino, Maslakova and Ventura, 2006). ^[6]
• Notably, there are non-CAT(0) free-by-cyclic groups (Gersten, 1994). ^[7]
• However, all free-by-cyclic groups satisfy a quadratic isoperimetric inequality (Bridson and Groves, 2010). ^[8]
• All free-by-cyclic groups where the underlying free group has rank ${\displaystyle 2}$ are CAT(0) (Brady, 1995). ^[9]
• Many examples of free-by-cyclic groups with polynomially-growing attaching maps are known to be CAT(0). ^{[10] [11]}
• Free-by-cyclic groups are equationally noetherian and have well-ordered growth rates (Kudlinska, Valiunas, 2024 preprint). ^[12]

References

1. Brinkmann, P. (2000-12-01). "Hyperbolic automorphisms of free groups". Geometric and Functional Analysis. 10 (5): 1071–1089. arXiv: math/9906008 . 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. arXiv: 1406.3292 . 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". Memoirs of the American Mathematical Society. 203 (955). arXiv: math/0610332 . doi:10.1090/S0065-9266-09-00578-X . 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. Kudlinska, Monika; Valiunas, Motiejus (2024). "Free-by-cyclic groups are equationally Noetherian". arXiv: 2407.08809 [math.GR].