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]