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 $K=\langle a,tb\rangle \leq G$, 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 $n\geq 2$, 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 $n\geq 1$ 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 $n\geq 4$ 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 $n\geq |r|$ 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 $m\geq 2,n\geq 1,d\geq 1,$ 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 $H,K\leq G$ 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 $\mathbb {Z} \ wr\ \mathbb {Z}$.

See also

• Hanna Neumann conjecture