Howson property

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Short description: Mathematical 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 [math]\displaystyle{ G }[/math] is said to have the Howson property if for every finitely generated subgroups [math]\displaystyle{ H,K }[/math] of [math]\displaystyle{ G }[/math] their intersection [math]\displaystyle{ H\cap K }[/math] is again a finitely generated subgroup of [math]\displaystyle{ G }[/math].[2]

Examples and non-examples

  • Every finite group has the Howson property.
  • The group [math]\displaystyle{ G=F(a,b)\times \mathbb Z }[/math] does not have the Howson property. Specifically, if [math]\displaystyle{ t }[/math] is the generator of the [math]\displaystyle{ \mathbb Z }[/math] factor of [math]\displaystyle{ G }[/math], then for [math]\displaystyle{ H=F(a,b) }[/math] and [math]\displaystyle{ K=\langle a,tb\rangle \le G }[/math], one has [math]\displaystyle{ H\cap K=\operatorname{ncl}_{F(a,b)}(a) }[/math]. Therefore, [math]\displaystyle{ H\cap K }[/math] is not finitely generated.[3]
  • If [math]\displaystyle{ \Sigma }[/math] is a compact surface then the fundamental group [math]\displaystyle{ \pi_1(\Sigma) }[/math] of [math]\displaystyle{ \Sigma }[/math] has the Howson property.[4]
  • A free-by-(infinite cyclic group) [math]\displaystyle{ F_n\rtimes \mathbb Z }[/math], where [math]\displaystyle{ n\ge 2 }[/math], 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 [math]\displaystyle{ \pi_1(M) }[/math] 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 [math]\displaystyle{ n\ge 1 }[/math] the Baumslag–Solitar group [math]\displaystyle{ BS(1,n)=\langle a,t\mid t^{-1}at=a^n\rangle }[/math] 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 [math]\displaystyle{ A,B }[/math] are groups with the Howson property then their free product [math]\displaystyle{ A\ast B }[/math] 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 [math]\displaystyle{ F,F' }[/math] and an infinite cyclic group [math]\displaystyle{ C }[/math], the amalgamated free product [math]\displaystyle{ F\ast_C F' }[/math] has the Howson property if and only if [math]\displaystyle{ C }[/math] is a maximal cyclic subgroup in both [math]\displaystyle{ F }[/math] and [math]\displaystyle{ F' }[/math].[10]
  • A right-angled Artin group [math]\displaystyle{ A(\Gamma) }[/math] has the Howson property if and only if every connected component of [math]\displaystyle{ \Gamma }[/math] is a complete graph.[11]
  • Limit groups have the Howson property.[12]
  • It is not known whether [math]\displaystyle{ SL(3,\mathbb Z) }[/math] has the Howson property.[13]
  • For [math]\displaystyle{ n\ge 4 }[/math] the group [math]\displaystyle{ SL(n,\mathbb Z) }[/math] contains a subgroup isomorphic to [math]\displaystyle{ F(a,b)\times F(a,b) }[/math] 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 [math]\displaystyle{ G=\langle x_1,\dots, x_k \mid r^n=1\rangle }[/math], where [math]\displaystyle{ n\ge |r| }[/math] 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 [math]\displaystyle{ F }[/math] satisfies a topological version of the Howson property: If [math]\displaystyle{ H,K }[/math] are topologically finitely generated closed subgroups of [math]\displaystyle{ F }[/math] then their intersection [math]\displaystyle{ H\cap K }[/math] is topologically finitely generated.[19]
  • For any fixed integers [math]\displaystyle{ m\ge 2,n\ge 1,d\ge 1, }[/math] a ``generic" [math]\displaystyle{ m }[/math]-generator [math]\displaystyle{ n }[/math]-relator group [math]\displaystyle{ G=\langle x_1,\dots x_m|r_1,\dots, r_n\rangle }[/math] has the property that for any [math]\displaystyle{ d }[/math]-generated subgroups [math]\displaystyle{ H,K\le G }[/math] their intersection [math]\displaystyle{ H\cap K }[/math] is again finitely generated.[20]
  • The wreath product [math]\displaystyle{ \mathbb Z\ wr\ \mathbb Z }[/math] does not have the Howson property.[21]
  • Thompson's group [math]\displaystyle{ F }[/math] does not have the Howson property, since it contains [math]\displaystyle{ \mathbb Z\ wr\ \mathbb Z }[/math].[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. ↑ 3.0 3.1 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. ↑ 6.0 6.1 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. ↑ 13.0 13.1 D. D. Long and A. W. Reid, Small Subgroups of [math]\displaystyle{ SL(3, \mathbb Z) }[/math], 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 [math]\displaystyle{ F }[/math] and other diagram groups. Sbornik: Mathematics 190.8 (1999): 1077-1130; Corollary 20.



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