{
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  "body": "\n## Summary\n\nIn 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|020Z5SEN27]] [[normal subgroup|0CB0DX5BVV]] $F$ such that the [[quotient group|0KYX2N69J0]] $G/F$ is [[cyclic|0B85Q41RP6]]. 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.\n",
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