Properties of groups | Infinite group theory

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. (Wikipedia).

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Geometric group theory | Group extension | Normal subgroup | Quotient group | Group theory | Hyperbolic group | Relatively hyperbolic group | Cyclic group | Free group | Group (mathematics)