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.