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 [math]\displaystyle{ G }[/math] is said to be free-by-cyclic if it has a free normal subgroup [math]\displaystyle{ F }[/math] such that the quotient group [math]\displaystyle{ G/F }[/math] is cyclic. In other words, [math]\displaystyle{ G }[/math] 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 [math]\displaystyle{ F }[/math] is finitely generated and the quotient is an infinite cyclic group. Equivalently, we can define a free-by-cyclic group constructively: if [math]\displaystyle{ \varphi }[/math] is an automorphism of [math]\displaystyle{ F }[/math], the semidirect product [math]\displaystyle{ F \rtimes_\varphi \mathbb{Z} }[/math] is a free-by-cyclic group.

An isomorphism class of a free-by-cyclic group is determined by an outer automorphism. If two automorphisms [math]\displaystyle{ \varphi, \psi }[/math] represent the same outer automorphism, that is, [math]\displaystyle{ \varphi = \psi\iota }[/math] for some inner automorphism [math]\displaystyle{ \iota }[/math], the free-by-cyclic groups [math]\displaystyle{ F \rtimes_\varphi \mathbb{Z} }[/math] and [math]\displaystyle{ F \rtimes_\psi \mathbb{Z} }[/math] 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 [math]\displaystyle{ \mathbb{Z}^2 }[/math], 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 [math]\displaystyle{ 2 }[/math] 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" (in en). Geometric and Functional Analysis 10 (5): 1071–1089. doi:10.1007/PL00001647. ISSN 1420-8970. https://link.springer.com/article/10.1007/PL00001647. 
  2. ↑ Hagen, Mark F.; Wise, Daniel T. (2015-02-01). "Cubulating hyperbolic free-by-cyclic groups: the general case" (in en). Geometric and Functional Analysis 25 (1): 134–179. doi:10.1007/s00039-015-0314-y. ISSN 1420-8970. https://link.springer.com/article/10.1007/s00039-015-0314-y. 
  3. ↑ Ghosh, Pritam (2023). "Relative hyperbolicity of free-by-cyclic extensions" (in en). Compositio Mathematica 159 (1): 153–183. doi:10.1112/S0010437X22007813. ISSN 0010-437X. https://www.cambridge.org/core/journals/compositio-mathematica/article/relative-hyperbolicity-of-freebycyclic-extensions/B5F9111E9EBE0CE289D7B115CF61C2A0. 
  4. ↑ Dahmani, François; Li, Ruoyu (2022). "Relative hyperbolicity for automorphisms of free products and free groups" (in en). Journal of Topology and Analysis 14 (1): 55–92. doi:10.1142/S1793525321500011. ISSN 1793-5253. https://www.worldscientific.com/doi/10.1142/S1793525321500011. 
  5. ↑ Feighn, Mark; Handel, Michael (1999). "Mapping Tori of Free Group Automorphisms are Coherent". Annals of Mathematics 149 (3): 1061–1077. doi:10.2307/121081. ISSN 0003-486X. https://www.jstor.org/stable/121081. 
  6. ↑ Bogopolski, O.; Martino, A.; Maslakova, O.; Ventura, E. (2006). "The conjugacy problem is solvable in free-by-cyclic groups" (in en). Bulletin of the London Mathematical Society 38 (5): 787–794. doi:10.1112/S0024609306018674. ISSN 0024-6093. http://doi.wiley.com/10.1112/S0024609306018674. 
  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. https://www.jstor.org/stable/2161207. 
  8. ↑ Bridson, Martin; Groves, Daniel (2010). "The quadratic isoperimetric inequality for mapping tori of free group automorphisms" (in en). Memoirs of the American Mathematical Society 203 (955). doi:10.1090/S0065-9266-09-00578-X. http://www.ams.org/books/memo/0955/. 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. https://dx.doi.org/10.1016/0166-8641%2894%2900072-B. 
  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. https://www.sciencedirect.com/science/article/pii/S0166864105003214. 
  11. ↑ Lyman, Rylee Alanza (2023). "Some New CAT(0) Free-by-Cyclic Groups". Michigan Mathematical Journal 73 (3): 621–630. doi:10.1307/mmj/20205989. ISSN 0026-2285. https://projecteuclid.org/journals/michigan-mathematical-journal/volume-73/issue-3/Some-New-CAT0-Free-by-Cyclic-Groups/10.1307/mmj/20205989.short. 
  12. ↑ Kudlinska, Monika; Valiunas, Motiejus (2024). "Free-by-cyclic groups are equationally Noetherian". arXiv:2407.08809 [math.GR].



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