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 generatedsubgroups 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]
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]
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]
One-relator groups , where are also locally quasiconvex
word-hyperbolic groups and therefore have the Howson property.[16]
The
Grigorchuk groupG 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]
^O. Bogopolski,
Introduction to group theory.
Translated, revised and expanded from the 2002 Russian original. EMS Textbooks in Mathematics. European Mathematical Society (EMS), Zürich, 2008.
ISBN978-3-03719-041-8; p. 102
^
abD. I. Moldavanskii, The intersection of finitely generated subgroups(in Russian) Siberian Mathematical Journal 9 (1968), 1422–1426
^H. Servatius, C. Droms,
B. Servatius, The finite basis extension property and graph groups. Topology and combinatorial group theory (Hanover, NH, 1986/1987; Enfield, NH, 1988), 52–58,
Lecture Notes in Math., 1440, Springer, Berlin, 1990
^J. P. McCammond, D. T. Wise, Coherence, local quasiconvexity, and the perimeter of 2-complexes.
Geometric and Functional Analysis15 (2005), no. 4, 859–927
^A. V. Rozhkov,
Centralizers of elements in a group of tree automorphisms. (in Russian)
Izv. Ross. Akad. Nauk Ser. Mat. 57 (1993), no. 6, 82–105; translation in:
Russian Acad. Sci. Izv. Math. 43 (1993), no. 3, 471–492
^L. Ribes, and P. Zalesskii, Profinite groups. Second edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 40. Springer-Verlag, Berlin, 2010.
ISBN978-3-642-01641-7; Theorem 9.1.20 on p. 366
^G. N. Arzhantseva,
Generic properties of finitely presented groups and Howson's theorem.
Communications in Algebra26 (1998), no. 11, 3783–3792
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 generatedsubgroups 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]
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]
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]
One-relator groups , where are also locally quasiconvex
word-hyperbolic groups and therefore have the Howson property.[16]
The
Grigorchuk groupG 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]
^O. Bogopolski,
Introduction to group theory.
Translated, revised and expanded from the 2002 Russian original. EMS Textbooks in Mathematics. European Mathematical Society (EMS), Zürich, 2008.
ISBN978-3-03719-041-8; p. 102
^
abD. I. Moldavanskii, The intersection of finitely generated subgroups(in Russian) Siberian Mathematical Journal 9 (1968), 1422–1426
^H. Servatius, C. Droms,
B. Servatius, The finite basis extension property and graph groups. Topology and combinatorial group theory (Hanover, NH, 1986/1987; Enfield, NH, 1988), 52–58,
Lecture Notes in Math., 1440, Springer, Berlin, 1990
^J. P. McCammond, D. T. Wise, Coherence, local quasiconvexity, and the perimeter of 2-complexes.
Geometric and Functional Analysis15 (2005), no. 4, 859–927
^A. V. Rozhkov,
Centralizers of elements in a group of tree automorphisms. (in Russian)
Izv. Ross. Akad. Nauk Ser. Mat. 57 (1993), no. 6, 82–105; translation in:
Russian Acad. Sci. Izv. Math. 43 (1993), no. 3, 471–492
^L. Ribes, and P. Zalesskii, Profinite groups. Second edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 40. Springer-Verlag, Berlin, 2010.
ISBN978-3-642-01641-7; Theorem 9.1.20 on p. 366
^G. N. Arzhantseva,
Generic properties of finitely presented groups and Howson's theorem.
Communications in Algebra26 (1998), no. 11, 3783–3792