Free-by-cyclic group
In group theory, especially, in 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. 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 and results
The study of free-by-cyclic groups is strongly related to that of the attaching outer automorphism. Among the motivating questions are those concerning their non-positive curvature properties, such as being CAT(0).- A free-by-cyclic group is hyperbolic, if and only if it does not contain a subgroup isomorphic to, if and only if no nontrivial conjugacy class is left invariant by the attaching automorphism.
- Hyperbolic free-by-cyclic groups are fundamental groups of compact non-positively curved cube complexes.
- Some free-by-cyclic groups are [relatively hyperbolic group|hyperbolic group|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.
- Any finitely generated subgroup of a free-by-cyclic group is finitely presented.
- The conjugacy problem for free-by-cyclic groups is solved.
- Notably, there are non-CAT free-by-cyclic groups.
- However, all free-by-cyclic groups satisfy a quadratic isoperimetric inequality.
- All free-by-cyclic groups where the underlying free group has rank are CAT.
- Many examples of free-by-cyclic groups with polynomially-growing attaching maps are known to be CAT.
- Free-by-cyclic groups are equationally noetherian and have well-ordered growth rates.