In the mathematical subject of group theory, the Howson property, also known as the finitely generated intersection property , 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.
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.
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.
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.
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.
One-relator groups, where are also locally quasiconvex word-hyperbolic groups and therefore have the Howson property.
The Grigorchuk groupG 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.
