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

The class of free-by-cyclic groups contains various groups as follow:

• A free-by-cyclic group is hyperbolic if and only if the attaching map is atoroidal.
• 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.
• Notably, there is a non-CAT(0) free-by-cyclic group.

References

• A. Martino and E. Ventura (2004), The Conjugacy Problem for Free-by-Cyclic Groups Archived 2007-09-27 at the Wayback Machine. Preprint from the Centre de Recerca Matemàtica, Barcelona, Catalonia, Spain.

• Feighn, Mark; Handel, Michael Mapping tori of free group automorphisms are coherent, Ann. Math., Volume 149 (1999) no. 3

• Ghosh, P. (2023). Relative hyperbolicity of free-by-cyclic extensions. Compositio Mathematica, 159(1), 153-183.

• F. Dahmani and R. Li, Relative hyperbolicity for automorphisms of free products and free groups, Journal of Topology and AnalysisVol. 14, No. 01, pp. 55-92 (2022)