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  is said to be free-by-cyclic if it has a free normal subgroup  such that the quotient group  is cyclic. In other words,  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  is finitely generated and the quotient is an infinite cyclic group. Equivalently, we can define a free-by-cyclic group constructively: if  is an automorphism of , the semidirect product  is a free-by-cyclic group. An isomorphism class of a free-by-cyclic group is determined by an outer automorphism. If two automorphisms  represent the same outer automorphism, that is,  for some inner automorphism , the free-by-cyclic groups  and  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 . 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)

Infinite group theory
Properties of groups