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.A. G. Howson, On the intersection of finitely generated free groups. Journal of the London Mathematical Society 29 (1954), 428–434
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 .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. ; p. 102
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.D. I. Moldavanskii, The intersection of finitely generated subgroups Siberian Mathematical Journal 9 (1968), 1422–1426 If is a compact surface then the fundamental group of has the Howson property.L. Greenberg, Discrete groups of motions. Canadian Journal of Mathematics 12 (1960), 415–426 A free-by-(infinite cyclic group) , where , never has the Howson property.R. G. Burns and A. M. Brunner, Two remarks on the group property of Howson, Algebra i Logika 18 (1979), 513–522 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.T. Soma, 3-manifold groups with the finitely generated intersection property, Transactions of the American Mathematical Society, 331 (1992), no. 2, 761–769 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. For every the Baumslag–Solitar group has the Howson property. 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.V. Araújo, P. Silva, M. Sykiotis, Finiteness results for subgroups of finite extensions. Journal of Algebra 423 (2015), 592–614 If are groups with the Howson property then their free product also has the Howson property.B. Baumslag, Intersections of finitely generated subgroups in free products. Journal of the London Mathematical Society 41 (1966), 673–679 More generally, the Howson property is preserved under taking amalgamated free products and HNN-extension of groups with the Howson property over finite subgroups.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 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 .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 A right-angled Artin group has the Howson property if and only if every connected component of is a complete graph.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 Limit groups have the Howson property.F. Dahmani, Combination of convergence groups. Geometry & Topology 7 (2003), 933–963 It is not known whether has the Howson property.D. D. Long and A. W. Reid, Small Subgroups of , Experimental Mathematics, 20(4):412–425, 2011 For the group contains a subgroup isomorphic to and does not have the Howson property. 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.J. P. McCammond, D. T. Wise, Coherence, local quasiconvexity, and the perimeter of 2-complexes. Geometric and Functional Analysis 15 (2005), no. 4, 859–927P. Schupp, Coxeter groups, 2-completion, perimeter reduction and subgroup separability, Geometriae Dedicata 96 (2003) 179–198 One-relator groups , where are also locally quasiconvex word-hyperbolic groups and therefore have the Howson property.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 The Grigorchuk group G of intermediate growth does not have the Howson property.A. V. Rozhkov, Centralizers of elements in a group of tree automorphisms. 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 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.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. ; Theorem 10.4.13 on p. 236 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.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. ; Theorem 9.1.20 on p. 366 For any fixed integers a ``generic" -generator -relator group has the property that for any -generated subgroups their intersection is again finitely generated.G. N. Arzhantseva, Generic properties of finitely presented groups and Howson's theorem. Communications in Algebra 26 (1998), no. 11, 3783–3792 The wreath product does not have the Howson property.A. S. Kirkinski, Intersections of finitely generated subgroups in metabelian groups. Algebra i Logika 20 (1981), no. 1, 37–54; Lemma 3. Thompson's group does not have the Howson property, since it contains .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.
See also Hanna Neumann conjecture
References