# 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.[1]

## 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 ${\displaystyle G}$.[2]

## 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.[3]
• If ${\displaystyle \Sigma }$ is a compact surface then the fundamental group ${\displaystyle \pi _{1}(\Sigma )}$ of ${\displaystyle \Sigma }$ has the Howson property.[4]
• A free-by-(infinite cyclic group) ${\displaystyle F_{n}\rtimes \mathbb {Z} }$, where ${\displaystyle n\geq 2}$, 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 ${\displaystyle \pi _{1}(M)}$ 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 ${\displaystyle n\geq 1}$ the Baumslag–Solitar group ${\displaystyle BS(1,n)=\langle a,t\mid t^{-1}at=a^{n}\rangle }$ has the Howson property.[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 ${\displaystyle A,B}$ are groups with the Howson property then their free product ${\displaystyle A\ast B}$ 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 ${\displaystyle F,F'}$ 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 ${\displaystyle F'}$.[10]
• 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.[11]
• Limit groups have the Howson property.[12]
• It is not known whether ${\displaystyle SL(3,\mathbb {Z} )}$ has the Howson property.[13]
• 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.[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 ${\displaystyle G=\langle x_{1},\dots ,x_{k}\mid r^{n}=1\rangle }$, where ${\displaystyle n\geq |r|}$ are also locally quasiconvex word-hyperbolic groups and therefore have the Howson property.[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 ${\displaystyle F}$ 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.[19]
• 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.[20]
• The wreath product ${\displaystyle \mathbb {Z} \ wr\ \mathbb {Z} }$ does not have the Howson property.[21]
• Thompson's group ${\displaystyle F}$ does not have the Howson property, since it contains ${\displaystyle \mathbb {Z} \ wr\ \mathbb {Z} }$.[22]

## References

1. ^ A. G. Howson, On the intersection of finitely generated free groups. Journal of the London Mathematical Society 29 (1954), 428–434
2. ^ 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
3. ^ a b D. I. Moldavanskii, The intersection of finitely generated subgroups (in Russian) Siberian Mathematical Journal 9 (1968), 1422–1426
4. ^ L. Greenberg, Discrete groups of motions. Canadian Journal of Mathematics 12 (1960), 415–426
5. ^ R. G. Burns and A. M. Brunner, Two remarks on the group property of Howson, Algebra i Logika 18 (1979), 513–522
6. ^ 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
7. ^ V. Araújo, P. Silva, M. Sykiotis, Finiteness results for subgroups of finite extensions. Journal of Algebra 423 (2015), 592–614
8. ^ B. Baumslag, Intersections of finitely generated subgroups in free products. Journal of the London Mathematical Society 41 (1966), 673–679
9. ^ 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
10. ^ 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
11. ^ 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
12. ^ F. Dahmani, Combination of convergence groups. Geometry & Topology 7 (2003), 933–963
13. ^ a b D. D. Long and A. W. Reid, Small Subgroups of ${\displaystyle SL(3,\mathbb {Z} )}$, Experimental Mathematics, 20(4):412–425, 2011
14. ^ 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
15. ^ P. Schupp, Coxeter groups, 2-completion, perimeter reduction and subgroup separability, Geometriae Dedicata 96 (2003) 179–198
16. ^ 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
17. ^ 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
18. ^ 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
19. ^ 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
20. ^ G. N. Arzhantseva, Generic properties of finitely presented groups and Howson's theorem. Communications in Algebra 26 (1998), no. 11, 3783–3792
21. ^ A. S. Kirkinski, Intersections of finitely generated subgroups in metabelian groups. Algebra i Logika 20 (1981), no. 1, 37–54; Lemma 3.
22. ^ V. Guba and M. Sapir, On subgroups of R. Thompson's group ${\displaystyle F}$ and other diagram groups. Sbornik: Mathematics 190.8 (1999): 1077-1130; Corollary 20.