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 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 (irreducible case: Bestvina and Feighn, 1992; general case: Brinkmann, 2000).
- Hyperbolic free-by-cyclic groups are fundamental groups of compact non-positively curved cube complexes (Hagen and Wise, 2015).
- 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.
- Any finitely generated subgroup of a free-by-cyclic group is finitely presented (Feighn and Handel, 1999).
- The conjugacy problem for free-by-cyclic groups is solved (Bogopolski, Martino, Maslakova and Ventura, 2006).
- Notably, there are non-CAT(0) free-by-cyclic groups (Gersten, 1994).
- However, all free-by-cyclic groups satisfy a quadratic isoperimetric inequality (Bridson and Groves, 2010).
- All free-by-cyclic groups where the underlying free group has rank are CAT(0) (Brady, 1995).
- Many examples of free-by-cyclic groups with polynomially-growing attaching maps are known to be CAT(0).
- Free-by-cyclic groups are equationally noetherian and have well-ordered growth rates (Kudlinska, Valiunas, 2024 preprint).
References