# On the intersection of finitely generated subgroups of free groups

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Robert G. Burns, ‘On the intersection of finitely generated subgroups of a free group’, Math. (), – MathSciNet CrossRef zbMATH Google ScholarCited by: We study the intersection of finitely generated subgroups of free groups by utilizing the method of linear programming.

We prove that if $$H_1$$ is a finitely generated subgroup of a free group F. A subgroup H of a free product of groups G α, α∈ I, is called factor free if for every and β ∈ I one has S H S-1 ∩ G β = {1} (by Kurosh theorem on subgroups of free products, factor free subgroups are free).

If K is a finitely generated free group, denote, where r(K) is the rank of is proven that if H, K are finitely generated factor free subgroups of a free product by: If H is a finitely generated subgroup of a free group F, then H is a free factor of a subgroup 7H of finite index in F.

In this paper we shall show how a number of results about finitely generated subgroups of a free group follow in a natural way from the above special case.

In fact, one of several ways to define a polycyclic group is to demand that it is a solvable group for which all subgroups are finitely generated. So, this might seem kind of tautological, but polycyclic groups have other definitions and they come up quite a bit in various areas.

generated subgroups of free groups to free products. He proved that a finitely generated normal subgroup (^ 1) of a free On the intersection of finitely generated subgroups of free groups book is of finite index; and if A, B have the finitely generated intersection property (i.e., the intersection of any two finitely generated subgroups is finitely generated), then G = A*B has the finitely.

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R.G. Burns, On the intersection of finitely generated subgroups of a free group, Math. (), – MathSciNet CrossRef zbMATH Google Scholar. Subgroups of a Cyclic Group. Intersection and Generators.

Finitely Generated Free Abelian Groups. Finitely Ge erated Abelian Groups. Invariants and Elementary Divisors. GENERATORS AND RELATIONS. Other editions - View all. Do finitely generated nilpotent groups contain torsion free subgroups of finite index.

Hot Network Questions Why does a flight from Melbourne to Cape Town need to go via Doha. Abstract. The aim of this paper is to present computing techniques for the finitely generated subgroups of the free group. The focus is the computation of the closures of a finitely generated subgroup for certain profinite topologies, namely the pro-p and the pro-nilpotent r the first section concentrates on the classical manipulation of finitely generated subgroups when more.

A group G is said to have the finitely generated intersection property (for short FGIP) if, for each pair of finitely generated subgroups H, K c G, HnK is finitely generated. Greenberg  proved that the fundamental groups of surfaces have FGIP. For given 3-manifolds M, we would like to know if their fundamental groups 7ii(M) have FGIP or not.

$\begingroup$ Is there a finitely generated subgroup of infinite index with maximal growth exponent. I suppose at the very least one could `cut off' $\Gamma_0$ (so $0\leq k\leq n$, say) and obtain a sequence of subgroups with exponent tending towards $\log 3$.

$\endgroup$ – HJRW Oct 28 at   [11, Chapter I, Theorem ] Let G be a finitely generated abelian group.

### Details On the intersection of finitely generated subgroups of free groups EPUB

Then the torsion subgroup G tor is finite, G G tor is free and there exists a free subgroup of H of G such that G is the direct sum of G tor and H. The rank of the free group H ≃ G G tor is also called the rank of G. Theorems – completely describe the structure.

On the Intersection of Finitely Generated Subgroups of Free Groups. (41 praved that the intersection of twafinitely gener- ated subgroups H and 1finitely. A group is residually finite if and only if the intersection of all its normal subgroups of finite index is trivial.

Examples of groups that are residually finite are finite groups, free groups, finitely generated nilpotent groups, polycyclic-by-finite groups, finitely generated linear groups, and fundamental groups of. Pub Date: July arXiv: arXiv Bibcode: arXivI Keywords: Mathematics - Group Theory; Computer Science - Discrete Mathematics.

Robert G. Burns, "On the intersection of finitely generated subgroups of a free group", Math. (), – MR43# MathSciNet CrossRef zbMATH Google Scholar. If H is a finitely generated group, then Γ〈h 1,h n 〉, the Cayley graph of H with respect to a finite generating set {h 1,h n}, has as vertices the elements of is an edge between the vertices v and w of Γ if vh i = w for some i ∈ {1,n}.

Theorem. Let H be a negatively curved group. If A is an infinite quasiconvex subgroup of H (i.e. there is a real number ε such. Abstract. The Fuchsian groups are the discrete subgroups of LF(2, R), the group of all 2 × 2 matrices over the reels with determinant + are interested here in the following group-theoretical property in particular in connection with Fuchsian groups: a group is said to have the finitely generated intersection property if the intersection of every pair of finitely generated subgroups is.

It is infinite and has a central extension $\tilde B$ described in Aryan's book "Burnside problems and identities of groups" such that the intersection of any two cyclic subgroups of $\tilde B$ is infinite. A subgroup H of a free product of groups G α, α∈ I, is termed factor free if for every and β∈I one has SHS-1 ∩G β = {1} (by Kurosh theorem on subgroups of free products, factor free subgroups are free).

If K is a finitely generated free group, denote, where r(K) is the rank of has earlier been proved by the author that if H, K are finitely generated factor free subgroups of then.

Abstract.

### Description On the intersection of finitely generated subgroups of free groups EPUB

We study the intersection of finitely generated factor-free subgroups of free products of groups by utilizing the method of linear programming. 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 property is named after Albert G.

Howson who in a paper established that free groups have this property. It is shown that ifH, Kare any finitely generated subgroups of a free groupFandUis any cyclic subgroup ofF, then any intersectionHg 1 U ∩ Kg 2 Uof double cosets contains only a finite number of double cosets (H ∩ K)gU, and an explicit upper bound for this number is given in terms of the ranks ofHandKand a generator result is then applied to the intersection of finitely generated.

The usual way to investigate the statistical properties of finitely generated subgroups of free groups, and of finite presentations of groups, is based on the so-called word-based distribution.

This paper has as its chief aim the establishment of two formulae associated with subgroups of finite index in free groups. The first of these (Theorem ) gives an expression for the total length of the free generators of a subgroup U of the free group F r with r generators.

In the mathematical subject of group theory, the Hanna Neumann conjecture is a statement about the rank of the intersection of two finitely generated subgroups of a free conjecture was posed by Hanna Neumann in Ina strengthened version of the conjecture (see below) was proved independently by Joel Friedman and by Igor Mineyev.

Ina third proof of the Strengthened. From this specification it can easily be determined whether the base is finite. In addi- tion, a polynomial-time algorithm to determine if a regular set is a circular code is presented. Introduction It is known  that the intersection of two finitely generated subgroups of a free group is again finitely generated.

We study the intersection of finitely generated subgroups of free groups by utilizing the method of linear programming. We prove that if H 1 is a finitely generated subgroup of a free group F, then the WN-coefficient σ(H 1) of H 1 is rational and can be computed in deterministic exponential time in the size of H coefficient σ(H 1) is the minimal nonnegative real number such that, for.

tions of nitely generated subgroups of metabelian groups as a whole are nitely generated. Here we shall prove that there are algorithms which decide for a free metabelian group (or, more generally, for the wreath product of two free abelian groups) whether the intersection of two nitely generated subgroups is nitely generated or trivial.

The. A subgroup of finite index in a finitely generated group is always finitely generated, and the Schreier index formula gives a bound on the number of generators required. InAlbert G. Howson showed that the intersection of two finitely generated subgroups of a free group is again finitely generated.

Here are some specific examples. Example (Infinitely generated subgroups of surface groups). Let G be a finitely generated free group or closed surface group which acts by isometries on the hyperbolic plane H 2 R, and let φ:G → Z be an epimorphism. It is easy to see that the kernel G ∗ of φ is not finitely generated.non abelian free groups or anything that surjects onto one of these groups.

Some other examples were given in  where it was shown that this property holds for any nitely generated LERF group (one where every nitely generated subgroup is the intersection of nite index subgroups) containing a non abelian free group.