Free-by-cyclic group explained

is said to be free-by-cyclic if it has a free normal subgroup such that the quotient group is cyclic. In other words, 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 is finitely generated and the quotient is an infinite cyclic group. Equivalently, we can define a free-by-cyclic group constructively: if is an automorphism of , the semidirect product is a free-by-cyclic group.

An isomorphism class of a free-by-cyclic group is determined by an outer automorphism. If two automorphisms

represent the same outer automorphism, that is, for some inner automorphism , the free-by-cyclic groups and 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

, 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]

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]

Notes and References

1. Brinkmann . P. . 2000-12-01 . Hyperbolic automorphisms of free groups . Geometric and Functional Analysis . en . 10 . 5 . 1071–1089 . 10.1007/PL00001647 . 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 . en . 25 . 1 . 134–179 . 10.1007/s00039-015-0314-y . 1420-8970.
3. Ghosh . Pritam . 2023 . Relative hyperbolicity of free-by-cyclic extensions . Compositio Mathematica . en . 159 . 1 . 153–183 . 10.1112/S0010437X22007813 . 0010-437X. 1802.08570 .
4. Dahmani . François . Li . Ruoyu . 2022 . Relative hyperbolicity for automorphisms of free products and free groups . Journal of Topology and Analysis . en . 14 . 1 . 55–92 . 10.1142/S1793525321500011 . 1793-5253. 1901.06760 .
5. Feighn . Mark . Handel . Michael . 1999 . Mapping Tori of Free Group Automorphisms are Coherent . Annals of Mathematics . 149 . 3 . 1061–1077 . 10.2307/121081 . 121081 . 0003-486X. math/9905209 .
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 . en . 38 . 5 . 787–794 . 10.1112/S0024609306018674 . 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 . 10.2307/2161207 . 2161207 . 0002-9939.
8. Web site: Bridson . Martin . Groves . Daniel . 2010 . The quadratic isoperimetric inequality for mapping tori of free group automorphisms . 2024-11-02 . American Mathematical Society . en.
9. Brady . Thomas . 1995-05-26 . Complexes of nonpositive curvature for extensions of F2 by Z . Topology and Its Applications . 63 . 3 . 267–275 . 10.1016/0166-8641(94)00072-B . 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 . 10.1016/j.topol.2005.12.002 . 0166-8641.
11. Lyman . Rylee Alanza . 2023 . Some New CAT(0) Free-by-Cyclic Groups . Michigan Mathematical Journal . 73 . 3 . 621–630 . 10.1307/mmj/20205989 . 0026-2285. 1909.03097 .
12. Web site: Free-by-cyclic groups are equationally Noetherian . 2024-11-02 . arxiv.org.