Howson property

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 $$G$$ is said to have the Howson property if for every finitely generated subgroups $$H,K$$ of $$G$$ their intersection $$H\cap K$$ is again a finitely generated subgroup of $$G$$.

Examples and non-examples

 * Every finite group has the Howson property.
 * The group $$G=F(a,b)\times \mathbb Z$$ does not have the Howson property. Specifically, if $$t$$ is the generator of the $$\mathbb Z$$ factor of $$G$$, then for $$H=F(a,b)$$ and $$K=\langle a,tb\rangle \le G $$, one has $$H\cap K=\operatorname{ncl}_{F(a,b)}(a)$$. Therefore, $$H\cap K$$ is not finitely generated.
 * If $$\Sigma$$ is a compact surface then the fundamental group $$\pi_1(\Sigma)$$ of $$\Sigma$$ has the Howson property.
 * A free-by-(infinite cyclic group) $$F_n\rtimes \mathbb Z$$, where $$n\ge 2$$, 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 $$\pi_1(M)$$ 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 $$n\ge 1$$ the Baumslag–Solitar group $$BS(1,n)=\langle a,t\mid t^{-1}at=a^n\rangle$$ 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 $$A,B$$ are groups with the Howson property then their free product $$A\ast B$$ 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 $$F,F'$$ and an infinite cyclic group $$C$$, the amalgamated free product $$F\ast_C F'$$ has the Howson property if and only if $$C$$ is a maximal cyclic subgroup in both $$F$$ and $$F'$$.
 * A right-angled Artin group $$A(\Gamma)$$ has the Howson property if and only if every connected component of $$\Gamma$$ is a complete graph.
 * Limit groups have the Howson property.
 * It is not known whether $$SL(3,\mathbb Z)$$ has the Howson property.
 * For $$n\ge 4$$ the group $$SL(n,\mathbb Z)$$ contains a subgroup isomorphic to $$F(a,b)\times F(a,b)$$ 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 $$G=\langle x_1,\dots, x_k \mid r^n=1\rangle$$, where $$n\ge |r|$$ 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 $$F$$ satisfies a topological version of the Howson property: If $$H,K$$ are topologically finitely generated closed subgroups of $$F$$ then their intersection $$H\cap K$$ is topologically finitely generated.
 * For any fixed integers $$m\ge 2,n\ge 1,d\ge 1,$$ a ``generic" $$m$$-generator $$n$$-relator group $$G=\langle x_1,\dots x_m|r_1,\dots, r_n\rangle$$ has the property that for any $$d$$-generated subgroups $$H,K\le G$$ their intersection $$H\cap K$$ is again finitely generated.
 * The wreath product $$\mathbb Z\ wr\ \mathbb Z$$ does not have the Howson property.
 * Thompson's group $$F$$ does not have the Howson property, since it contains $$\mathbb Z\ wr\ \mathbb Z$$.