# Howson property

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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[edit]

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[edit]

- 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[edit]

## References[edit]

**^**A. G. Howson,*On the intersection of finitely generated free groups*. Journal of the London Mathematical Society**29**(1954), 428–434**^**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. ISBN 978-3-03719-041-8; p. 102- ^
^{a}^{b}D. I. Moldavanskii,*The intersection of finitely generated subgroups*(in Russian) Siberian Mathematical Journal**9**(1968), 1422–1426 **^**L. Greenberg,*Discrete groups of motions*. Canadian Journal of Mathematics**12**(1960), 415–426**^**R. G. Burns and A. M. Brunner,*Two remarks on the group property of Howson*, Algebra i Logika**18**(1979), 513–522- ^
^{a}^{b}T. Soma, 3-manifold groups with the finitely generated intersection property, Transactions of the American Mathematical Society,**331**(1992), no. 2, 761–769 **^**V. Araújo, P. Silva, M. Sykiotis,*Finiteness results for subgroups of finite extensions*. Journal of Algebra**423**(2015), 592–614**^**B. Baumslag,*Intersections of finitely generated subgroups in free products*. Journal of the London Mathematical Society**41**(1966), 673–679**^**D. E. Cohen,*Finitely generated subgroups of amalgamated free products and HNN groups*. J. Austral. Math. Soc. Ser. A**22**(1976), no. 3, 274–281**^**R. G. Burns,*On the finitely generated subgroups of an amalgamated product of two groups*. Transactions of the American Mathematical Society**169**(1972), 293–306**^**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**^**F. Dahmani,*Combination of convergence groups*. Geometry & Topology**7**(2003), 933–963- ^
^{a}^{b}D. D. Long and A. W. Reid, Small Subgroups of , Experimental Mathematics, 20(4):412–425, 2011 **^**J. P. McCammond, D. T. Wise,*Coherence, local quasiconvexity, and the perimeter of 2-complexes*. Geometric and Functional Analysis**15**(2005), no. 4, 859–927**^**P. Schupp,*Coxeter groups, 2-completion, perimeter reduction and subgroup separability*, Geometriae Dedicata**96**(2003) 179–198**^**G. Ch. Hruska, D. T. Wise,*Towers, ladders and the B. B. Newman spelling theorem*. Journal of the Australian Mathematical Society**71**(2001), no. 1, 53–69**^**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**^**B. Fine, A. Gaglione, A. Myasnikov, G. Rosenberger, D. Spellman,*The elementary theory of groups. A guide through the proofs of the Tarski conjectures.*De Gruyter Expositions in Mathematics, 60. De Gruyter, Berlin, 2014. ISBN 978-3-11-034199-7; Theorem 10.4.13 on p. 236**^**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. ISBN 978-3-642-01641-7; Theorem 9.1.20 on p. 366**^**G. N. Arzhantseva,*Generic properties of finitely presented groups and Howson's theorem*. Communications in Algebra**26**(1998), no. 11, 3783–3792**^**A. S. Kirkinski,*Intersections of finitely generated subgroups in metabelian groups*. Algebra i Logika**20**(1981), no. 1, 37–54; Lemma 3.**^**V. Guba and M. Sapir,*On subgroups of R. Thompson's group and other diagram groups*. Sbornik: Mathematics**190.8**(1999): 1077-1130; Corollary 20.