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

An isomorphism class of a free-by-cyclic group is determined by an outer automorphism. If two automorphisms $\phi ,\psi$ represent the same outer automorphism, that is, $\phi =\psi \iota$ for some inner automorphism $\iota$, the free-by-cyclic groups $F{⋊}_{\phi }\mathbb{Z}$ and $F{⋊}_{\psi }\mathbb{Z}$ 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 ${\mathbb{Z}}^2$, 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). (Bridson and Groves 2010)

• Free-by-cyclic groups are conjugacy separable (Dahmani, Hughes, Kudlinska, and Touikan 2026)

• 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 $2$ 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 and Valiunas, 2025).