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]

A group $G$ is said to have the Howson property if for every finitely generated subgroups $H,K$ of $G$ their intersection $H\cap K$ is again a finitely generated subgroup of $G$ .^[2]

Examples and non-examples

Every finite group has the Howson property.
The group $G=F(a,b)\times \mathbb {Z}$ does not have the Howson property. Specifically, if $t$ is the generator of the $\mathbb {Z}$ factor of $G$ , then for $H=F(a,b)$ and $K=\langle a,tb\rangle \leq G$ , one has $H\cap K=\operatorname {ncl} _{F(a,b)}(a)$ . Therefore, $H\cap K$ is not finitely generated.^[3]
If $\Sigma$ is a compact surface then the fundamental group $\pi _{1}(\Sigma )$ of $\Sigma$ has the Howson property.^[4]
A free-by-(infinite cyclic group) $F_{n}\rtimes \mathbb {Z}$ , where $n\geq 2$ , 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 $\pi _{1}(M)$ 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 $n\geq 1$ the Baumslag–Solitar group $BS(1,n)=\langle a,t\mid t^{-1}at=a^{n}\rangle$ 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 $A,B$ are groups with the Howson property then their free product $A\ast B$ 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 $F,F'$ and an infinite cyclic group $C$ , the amalgamated free product $F\ast _{C}F'$ has the Howson property if and only if $C$ is a maximal cyclic subgroup in both $F$ and $F'$ .^[10]
A right-angled Artin group $A(\Gamma )$ has the Howson property if and only if every connected component of $\Gamma$ is a complete graph.^[11]
Limit groups have the Howson property.^[12]
It is not known whether $SL(3,\mathbb {Z} )$ has the Howson property.^[13]
For $n\geq 4$ the group $SL(n,\mathbb {Z} )$ contains a subgroup isomorphic to $F(a,b)\times F(a,b)$ 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.^[14]^[15]
One-relator groups $G=\langle x_{1},\dots ,x_{k}\mid r^{n}=1\rangle$ , where $n\geq |r|$ 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 $F$ satisfies a topological version of the Howson property: If $H,K$ are topologically finitely generated closed subgroups of $F$ then their intersection $H\cap K$ is topologically finitely generated.^[19]
For any fixed integers $m\geq 2,n\geq 1,d\geq 1,$ a ``generic" $m$ -generator $n$ -relator group $G=\langle x_{1},\dots x_{m}|r_{1},\dots ,r_{n}\rangle$ has the property that for any $d$ -generated subgroups $H,K\leq G$ their intersection $H\cap K$ is again finitely generated.^[20]
The wreath product $\mathbb {Z} \ wr\ \mathbb {Z}$ does not have the Howson property.^[21]
Thompson's group $F$ does not have the Howson property, since it contains $\mathbb {Z} \ wr\ \mathbb {Z}$ .^[22]

References

↑ A. G. Howson, On the intersection of finitely generated free groups. Journal of the London Mathematical Society 29 (1954), 428–434
↑ 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
1 2 D. I. Moldavanskii, The intersection of finitely generated subgroups (in Russian) Siberian Mathematical Journal 9 (1968), 1422–1426
↑ L. Greenberg, Discrete groups of motions. Canadian Journal of Mathematics 12 (1960), 415–426
↑ R. G. Burns and A. M. Brunner, Two remarks on the group property of Howson, Algebra i Logika 18 (1979), 513–522
1 2 T. Soma, 3-manifold groups with the finitely generated intersection property, Transactions of the American Mathematical Society, 331 (1992), no. 2, 761–769
↑ V. Araújo, P. Silva, M. Sykiotis, Finiteness results for subgroups of finite extensions. Journal of Algebra 423 (2015), 592–614
↑ B. Baumslag, Intersections of finitely generated subgroups in free products. Journal of the London Mathematical Society 41 (1966), 673–679
↑ 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
↑ 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
↑ 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
↑ F. Dahmani, Combination of convergence groups. Geometry & Topology 7 (2003), 933–963
1 2 D. D. Long and A. W. Reid, Small Subgroups of $SL(3,\mathbb {Z} )$ , Experimental Mathematics, 20(4):412–425, 2011
↑ 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
↑ P. Schupp, Coxeter groups, 2-completion, perimeter reduction and subgroup separability, Geometriae Dedicata 96 (2003) 179–198
↑ 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
↑ 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
↑ 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
↑ 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
↑ G. N. Arzhantseva, Generic properties of finitely presented groups and Howson's theorem. Communications in Algebra 26 (1998), no. 11, 3783–3792
↑ A. S. Kirkinski, Intersections of finitely generated subgroups in metabelian groups. Algebra i Logika 20 (1981), no. 1, 37–54; Lemma 3.
↑ V. Guba and M. Sapir, On subgroups of R. Thompson's group $F$ and other diagram groups. Sbornik: Mathematics 190.8 (1999): 1077-1130; Corollary 20.

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