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 $$G$$ 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.

An isomorphism class of a free-by-cyclic group is determined by an outer automorphism. If two automorphisms $$ \varphi, \psi $$ represent the same outer automorphism, that is, $$ \varphi = \psi\iota $$ for some inner automorphism $$ \iota $$, the free-by-cyclic groups $$ F \rtimes_\varphi \mathbb{Z} $$ and $$ F \rtimes_\psi \mathbb{Z} $$ 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.