Howson property

A group $G$ is said to have the Howson property if for every finitely generated subgroups $H,K$ of $G$ their intersection $H\backslash cap\; K$ is again a finitely generated subgroup of $G$.O. Bogopolski,

[https://books.google.com/books?id=jEw8MpP6DIgC&dq=Howson+property&pg=PA103 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

• Every finite group has the Howson property.
• The group $G=F(a,b)\backslash times\; \backslash mathbb\; Z$ does not have the Howson property. Specifically, if $t$ is the generator of the $\backslash mathbb\; Z$ factor of $G$, then for $H=F(a,b)$ and $K=\backslash langle\; a,tb\backslash rangle\; \backslash le\; G$, one has $H\backslash cap\; K=\backslash operatorname\{ncl\}\_\{F(a,b)\}(a)$. Therefore, $H\backslash cap\; K$ is not finitely generated.D. I. Moldavanskii, The intersection of finitely generated subgroups {{in lang|ru}} Siberian Mathematical Journal 9 (1968), 1422–1426
• If $\backslash Sigma$ is a compact surface then the fundamental group $\backslash pi\_1(\backslash Sigma)$ of $\backslash Sigma$ has the Howson property.L. Greenberg, Discrete groups of motions.

Canadian Journal of Mathematics 12 (1960), 415–426

• A free-by-(infinite cyclic group) $F\_n\backslash rtimes\; \backslash mathbb\; Z$, where $n\backslash ge\; 2$, 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 $\backslash pi\_1(M)$ does not have the Howson property.T. Soma, [https://www.ams.org/journals/tran/1992-331-02/S0002-9947-1992-1042289-4/S0002-9947-1992-1042289-4.pdf 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 $n\backslash ge\; 1$ the Baumslag–Solitar group $BS(1,n)=\backslash langle\; a,t\backslash mid\; t^\{-1\}at=a^n\backslash rangle$ 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 $A,B$ are groups with the Howson property then their free product $A\backslash ast\; B$ 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 $F,F\text{'}$ and an infinite cyclic group $C$, the amalgamated free product $F\backslash ast\_C\; F\text{'}$ has the Howson property if and only if $C$ is a maximal cyclic subgroup in both $F$ and $F\text{'}$.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 $A(\backslash Gamma)$ has the Howson property if and only if every connected component of $\backslash Gamma$ 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, [https://archive.org/details/arxiv-math0203258 Combination of convergence groups.] Geometry & Topology 7 (2003), 933–963
• It is not known whether $SL(3,\backslash mathbb\; Z)$ has the Howson property.D. D. Long and A. W. Reid, [https://projecteuclid.org/download/pdf_1/euclid.em/1323367155 Small Subgroups of $SL(3,\; \backslash mathbb\; Z)$], Experimental Mathematics, 20(4):412–425, 2011
• For $n\backslash ge\; 4$ the group $SL(n,\backslash mathbb\; Z)$ contains a subgroup isomorphic to $F(a,b)\backslash times\; F(a,b)$ 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 $G=\backslash langle\; x\_1,\backslash dots,\; x\_k\; \backslash mid\; r^n=1\backslash rangle$, where $n\backslash ge\; |r|$ 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. {{in lang|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

• 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, [https://books.google.com/books?id=pgbpBQAAQBAJ&dq=%22Howson+property%22&pg=PA236 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
• A free pro-p group $F$ satisfies a topological version of the Howson property: If $H,K$ are topologically finitely generated closed subgroups of $F$ then their intersection $H\backslash cap\; K$ 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. {{ISBN|978-3-642-01641-7}}; Theorem 9.1.20 on p. 366
• For any fixed integers $m\backslash ge\; 2,n\backslash ge\; 1,d\backslash ge\; 1,$ a ``generic" $m$-generator $n$-relator group $G=\backslash langle\; x\_1,\backslash dots\; x\_m|r\_1,\backslash dots,\; r\_n\backslash rangle$ has the property that for any $d$-generated subgroups $H,K\backslash le\; G$ their intersection $H\backslash cap\; K$ 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 $\backslash mathbb\; Z\backslash \; wr\backslash \; \backslash mathbb\; Z$ does not have the Howson property.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.

• Thompson's group $F$ does not have the Howson property, since it contains $\backslash mathbb\; Z\backslash \; wr\backslash \; \backslash mathbb\; Z$.V. Guba and M. Sapir,

[https://doi.org/10.1070/SM1999v190n08ABEH000419 On subgroups of R. Thompson's group $F$ and other diagram groups.]

Sbornik: Mathematics 190.8 (1999): 1077-1130; Corollary 20.

• Hanna Neumann conjecture

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Category:Group theory