In the mathematical subject of group theory, the Howson property, also known as the finitely generated intersection property (FGIP), is the property of a group saying that the intersection of any two finitely generated subgroups of this group is again finitely generated. The property is named after Albert G. Howson who in a 1954 paper established that free groups have this property.
Formal definition
A group is said to have the Howson property if for every finitely generated subgroups of their intersection is again a finitely generated subgroup of .
Examples and non-examples
- Every finite group has the Howson property.
- The group does not have the Howson property. Specifically, if is the generator of the factor of , then for and , one has . Therefore, is not finitely generated.
- If is a compact surface then the fundamental group of has the Howson property.
- A free-by-(infinite cyclic group) , where , never has the Howson property.
- In view of the recent proof of the Virtually Haken conjecture and the Virtually fibered conjecture for 3-manifolds, previously established results imply that if M is a closed hyperbolic 3-manifold then does not have the Howson property.
- Among 3-manifold groups, there are many examples that do and do not have the Howson property. 3-manifold groups with the Howson property include fundamental groups of hyperbolic 3-manifolds of infinite volume, 3-manifold groups based on Sol and Nil geometries, as well as 3-manifold groups obtained by some connected sum and JSJ decomposition constructions.
- For every the BaumslagâÂÂSolitar group has the Howson property.
- If G is group where every finitely generated subgroup is Noetherian then G has the Howson property. In particular, all abelian groups and all nilpotent groups have the Howson property.
- Every polycyclic-by-finite group has the Howson property.
- If are groups with the Howson property then their free product also has the Howson property. More generally, the Howson property is preserved under taking amalgamated free products and HNN-extension of groups with the Howson property over finite subgroups.
- In general, the Howson property is rather sensitive to amalgamated products and HNN extensions over infinite subgroups. In particular, for free groups and an infinite cyclic group , the amalgamated free product has the Howson property if and only if is a maximal cyclic subgroup in both and .
- A right-angled Artin group has the Howson property if and only if every connected component of is a complete graph.
- Limit groups have the Howson property.
- It is not known whether has the Howson property.
- For the group contains a subgroup isomorphic to and does not have the Howson property.
- Many small cancellation groups and Coxeter groups, satisfying the "perimeter reduction" condition on their presentation, are locally quasiconvex word-hyperbolic groups and therefore have the Howson property.
- One-relator groups , where are also locally quasiconvex word-hyperbolic groups and therefore have the Howson property.
- The Grigorchuk group G of intermediate growth does not have the Howson property.
- The Howson property is not a first-order property, that is the Howson property cannot be characterized by a collection of first order group language formulas.
- A free pro-p group satisfies a topological version of the Howson property: If are topologically finitely generated closed subgroups of then their intersection is topologically finitely generated.
- For any fixed integers a ``generic" -generator -relator group has the property that for any -generated subgroups their intersection is again finitely generated.
- The wreath product does not have the Howson property.
- Thompson's group does not have the Howson property, since it contains .
See also
References