Howson property

Mathematical property From Wikipedia, the free encyclopedia

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.[1]

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 .[2]

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.[3]
  • If is a compact surface then the fundamental group of has the Howson property.[4]
  • A free-by-(infinite cyclic group) , where , never has the Howson property.[5]
  • 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.[6]
  • 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.[6]
  • For every the Baumslag–Solitar group has the Howson property.[3]
  • 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.[7]
  • If are groups with the Howson property then their free product also has the Howson property.[8] More generally, the Howson property is preserved under taking amalgamated free products and HNN-extension of groups with the Howson property over finite subgroups.[9]
  • 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 .[10]
  • A right-angled Artin group has the Howson property if and only if every connected component of is a complete graph.[11]
  • Limit groups have the Howson property.[12]
  • It is not known whether has the Howson property.[13]
  • For the group contains a subgroup isomorphic to and does not have the Howson property.[13]
  • 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.[14][15]
  • One-relator groups , where are also locally quasiconvex word-hyperbolic groups and therefore have the Howson property.[16]
  • The Grigorchuk group G of intermediate growth does not have the Howson property.[17]
  • 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.[18]
  • 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.[19]
  • For any fixed integers a ``generic" -generator -relator group has the property that for any -generated subgroups their intersection is again finitely generated.[20]
  • The wreath product does not have the Howson property.[21]
  • Thompson's group does not have the Howson property, since it contains .[22]

See also

References

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