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词条 Howson property
释义

  1. Formal definition

  2. Examples and non-examples

  3. See also

  4. References

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 generated subgroups 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]
  • 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]

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, [https://books.google.com/books?id=jEw8MpP6DIgC&pg=PA103&dq=Howson+property&hl=en&sa=X&ved=0ahUKEwjGg4TZjOHWAhWJ7IMKHRwcBqQQ6AEIKDAA#v=onepage&q=Howson%20property&f=false 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. ^1 D. I. Moldavanskii, The intersection of finitely generated subgroups {{icon ru}} 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. ^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, [https://archive.org/details/arxiv-math0203258 Combination of convergence groups.] Geometry & Topology 7 (2003), 933–963
13. ^1 D. D. Long and A. W. Reid, [https://projecteuclid.org/download/pdf_1/euclid.em/1323367155 Small Subgroups of ], 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. {{icon ru}}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, [https://books.google.com/books?id=pgbpBQAAQBAJ&pg=PA236&lpg=PA236&dq=%22Howson+property%22&source=bl&ots=h_NB7Xl1RE&sig=4gPSSIePYnDV-4qKaLgkR3x17-o&hl=en&sa=X&ved=0ahUKEwifvfX7n-PWAhWPwYMKHVufBUk4ChDoAQgzMAM#v=onepage&q=%22Howson%20property%22&f=false 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,[https://link.springer.com/article/10.1007/BF01669493 Intersections of finitely generated subgroups in metabelian groups.]Algebra i Logika 20 (1981), no. 1, 37–54; Lemma 3.

2 : Group theory|Algebra
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