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

Examples and non-examples

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

See also

• Hanna Neumann conjecture

References