Free-by-cyclic group

Summary

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 $\varphi$ is an automorphism of $F$, the semidirect product $F \rtimes_\varphi \mathbb{Z}$ is a free-by-cyclic group.