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.<sup>[14][15]</sup>
• 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.<sup>[16]</sup>
• The Grigorchuk group G of intermediate growth does not have the Howson property.<sup>[17]</sup>
• 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.<sup>[18]</sup>
• 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.<sup>[19]</sup>
• 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]

See also

• Hanna Neumann conjecture

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. 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. 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. 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.