On Finitely Generated Subgroups of Free Products (original) (raw)

1971, Journal of the Australian Mathematical Society

https://doi.org/10.1017/S1446788700009824

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Abstract

If H is a subgroup of a group G we shall say that G is H-residually finite if for every element g in G, outside H, there is a subgroup of finite index in G, containing H and still avoiding g. (Then, according to the usual definition, G is residually finite if it is E-residually finite, where E is the identity subgroup). Definitions of other terms used below may be found in § 2 or in [6].

On residually finite groups and their generalizations

Colloquium Mathematicum, 1999

The paper is concerned with the class of groups satisfying the finite embedding (FE) property. This is a generalization of residually finite groups. In [2] it was asked whether there exist FE-groups which are not residually finite. Here we present such examples. To do this, we construct a family of three-generator soluble FE-groups with torsion-free abelian factors. We study necessary and sufficient conditions for groups from this class to be residually finite. This answers the questions asked in [1] and [2].

On the near Frattini subgroups of amalgamated free products with residual properties

1997

Let A(G), •b(G), and K(G, H) represent the lower near Frattini subgroup of G: A *H B, the near Frattini subgroup of G, and the core of H in G respectively. We show that G is A-free if G is residually finite, A and B have compatible H-filters, and H contains no nontrivial normal subgroup of G. We prove that if G is residually finite, then A(G) _ • K(G, H), provided: (1) A and B have compatible H-filters; (2) A and B each satisfies a nontrivial identical relation not satisfied by the infinite dihedral group; (3) A and B each satisfies a nontrivial identical relation and both I A: H[

Residual P-Finiteness of Certain Generalized Free Products of Nilpotent Groups

Kyungpook mathematical journal, 2008

We show that free products of finitely generated and residually p-finite nilpotent groups, amalgamating p-closed central subgroups are residually p-finite. As a consequence, we are able to show that generalized free products of residually p-finite abelian groups are residually p-finite if the amalgamated subgroup is closed in the prop topology on each of the factors.

Residual properties of free groups

Journal of Algebra, 1993

listed above contribute to the cost of publication of this Journal, but they are not owners or publishers and have no responsibility for its content or policies. Mathematical papers intended for publication in the Pacific Journal of Mathematics should be in typed form or offset-reproduced, (not dittoed), double spaced with large margins. Underline Greek letters in red, German in green, and script in blue. The first paragraph or two must be capable of being used separately as a synopsis of the entire paper. The editorial "we" must not be used in the synopsis, and items of the bibliography should not be cited there unless absolutely necessary, in which case they must be identified by author and Journal, rather than by item number. Manuscripts, in duplicate if possible, may be sent to any one of the four editors. Please classify according to the scheme of Math. Rev. Index to Vol.

Subgroups of free pro- p-products

Mathematical Proceedings of the Cambridge Philosophical Society, 1987

The results of this paper were announced at the Kertesz-Szele Group Theory Colloquium, Debrecen, September 1985. Added in proof. D. Haran has recently obtained a description of the countably generated subgroups of free prop products using a new concept of free product. (Cf. On closed subgroups of free products of profinite groups, manuscript.

On the finitely generated subgroups of an amalgamated product of two groups

Transactions of the American Mathematical Society, 1972

Sufficient conditions are found for the free product G G of two groups A A and B B with an amalgamated subgroup U U to have the properties (1) that the intersection of each pair of finitely generated subgroups of G G is again finitely generated, and (2) that every finitely generated subgroup containing a nontrivial subnormal subgroup of G G has finite index in G G . The known results that Fuchsian groups and free products (under the obvious conditions on the factors) have properties (1) and (2) follow as instances of the main result.

Subgroups of fully residually free groups: algorithmic problems

Group Theory, Statistics, and Cryptography, 2004

In [16] we introduced graph-theoretic techniques for finitely generated subgroups of F Z[t] and solved effectively the membership problem in finitely generated fully residually free groups. In the present paper we prove that finitely generated fully residually free groups satisfy Howson property and show how one can effectively find the intersection of two finitely generated subgroups, we solve the conjugacy problem, the malnormality problem, and provide an algorithm to compute ranks of centralizers. Contents 36 6. Conjugacy problem in finitely generated subgroups of F Z[t] 38 References 39 1991 Mathematics Subject Classification. 20E08. Key words and phrases. fully residually free groups, algorithmic problems.

On Subgroups of Finite Index in Positively Finitely Generated Groups

Bulletin of the London Mathematical Society, 2005

This paper proves that a subgroup of finite index in a positively finitely generated profinite group has maximal subgroup growth at most n log(n). In particular such a subgroup cannot be free, answering a question by L. Pyber.

Fully Residually Free Groups and Graphs Labeled by Infinite Words

International Journal of Algebra and Computation, 2006

Let F = F(X) be a free group with basis X and ℤ[t] be a ring of polynomials with integer coefficients in t. In this paper we develop a theory of (ℤ[t],X)-graphs — a powerful tool in studying finitely generated fully residually free (limit) groups. This theory is based on the Kharlampovich–Myasnikov characterization of finitely generated fully residually free groups as subgroups of the Lyndon's group Fℤ[t], the author's representation of elements of Fℤ[t] by infinite (ℤ[t],X)-words, and Stallings folding method for subgroups of free groups. As an application, we solve the membership problem for finitely generated subgroups of Fℤ[t], as well as for finitely generated fully residually free groups.

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References (7)

  1. R. G. Burns, 'A note on free groups', Proc. Amer. Math. Soc, 23 (1969), 14-17.
  2. I. M. S. Dey, 'Schreier systems in free products', Proc. Glasgow Math. Assoc, 1 (1965-66), 61-79.
  3. K. W. Gruenberg, 'Residual properties of infinite soluble groups', Proc. London Math. Soc. (3) 7 (1957), 29-62.
  4. M. Hall, Jr., 'Coset representation in free groups', Trans. Amer. Math. Soc. 67 (1949), 421- 432.
  5. S. MacLane, 'A proof of the subgroup theorem for free products', Mathematika 5 (1958), 161-183.
  6. W. Magnus, A. Karrass and D. Solitar, Combinatorial group theory (Interscience, New York, 1966).
  7. A. Karrass and 0 . Solitar, 'On finitely generated subgroups of a free group', Proc. Amer. Math. Soc, 22 (1969), 209-213.

Residually finite properties of groups / Muhammad Sufi Mohd Asri

2018

In this thesis, we shall study two stronger forms of residual finiteness, namely cyclic subgroup separability and weak potency in various generalized free products and HNN extensions. Among our results, we shall show that the generalized free products and HNN extensions where the amalgamated or associated subgroups are finite, or central, or infinite cyclic, or they are direct products of an infinite cyclic subgroup with a finite subgroup, or they are finite extensions of central subgroups, are again cyclic subgroup separable or weakly potent respectively. In order to prove our results, we shall prove a criterion each for the weak potency of generalized free products and HNN extensions, but we shall use previously established criterions for cyclic subgroup separability. Finally, we shall extend our results to tree products and fundamental groups of graphs of groups.

Residual properties of free groups II

Bulletin of the Australian Mathematical Society, 1972

In this paper it is proved that non-abelian free groups are residually [x, y \ x =1, y n =l, x = y) if and only if min{(m, k), (n, h)} is greater than 1 , and not both of (m, k) and (n, h) are 2 (where 0 is taken as greater than any natural number). The proof makes use of a result, possibly of independent interest, concerning the existence of certain automorphisms of the free group of rank two. A useful criterion which enables one to prove that non-abelian free groups are residually G for a large number of groups G is also given.

Finite Index Subgroups of Fully Residually Free Groups

International Journal of Algebra and Computation, 2011

Using graph-theoretic techniques for f.g. subgroups of Fℤ[t] we provide a criterion for a f.g. subgroup of a f.g. fully residually free group to be of finite index. Moreover, we show that this criterion can be checked effectively. As an application we obtain an analogue of Greenberg–Stallings Theorem for f.g. fully residually free groups, and prove that a f.g. nonabelian subgroup of a f.g. fully residually free group is of finite index in its normalizer and commensurator.

The isomorphism problem for finitely generated fully residually free groups

Journal of Pure and Applied Algebra, 2007

We prove that the isomorphism problem for finitely generated fully residually free groups (or F-groups for short) is decidable. We also show that each freely indecomposable F-group G has a decomposition that is invariant under automorphisms of G, and obtain a structure theorem for the group of outer automorphisms Out(G).

Subgroups of amalgamated free products

Bulletin of the American Mathematical Society, 1970

In 1934 Kuros [S] proved that "a subgroup of a free product of groups is again a free product." Several attempts have been made to extend this to a result about a free product of groups with an amalgamated subgroup, notably [4] and [ô]. Theorem 1 here gives to any subgroup of a free product with amalgamated subgroup an induced structure of the same type. We here indicate very briefly the method of proof. Details and related results will appear elsewhere. DEFINITION 0. Let G> be groups, for /x in an index set M, and let G be a group which is isomorphic to a subgroup of each G M under given maps ô M : G-»G M . The free product of the groups G> with the amalgamated subgroup G, denoted G = ( * M G>) 0t is the factor group of the free product ( * M G M ) with respect to the normal subgroup generated by all elements of the form 5 fi (g)ô v (g)~1 f where g runs through G and the pair (ju, v) runs through MXM.

Bounding the residual finiteness of free groups

2016

We find a lower bound to the size of finite groups detecting a given word in the free group, more precisely we construct a word wn of length n in non-abelian free groups with the property that wn is the identity on all finite quotients of size ∼ n 2/3 or less. This improves on a previous result of Bou-Rabee and McReynolds quantifying the lower bound of the residual finiteness of free groups.

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Ends of pairs of groups

Journal of Pure and Applied Algebra, 1977

In [lo], Hopf gave a definition of the number of ends, e(G), of a finitely generated (f.g.) group G. In [ 131, Specker extended his definition to cover arbitrary 'groups. They showed that the function e(G) could only kake the values 0, 1,2 or 0~ and they characterised those groups with two ends. In [14], Stallings characterised those f.g. groups G with at least two ends. We will say that a group G splits over a subgroup C if either G is a HNN extension A *c or G is an amalgamated free product A *J3 with A # C# B. Then Stallings' result says that a f.g. group G has e(G)32 if and only if G splits over some finite subgroup. This paper arose from an attempt to generalise the above result to groups which split over infinite subgroups. There is a natural definition, due to Houghton [ 1 l], of the number of ends, e(G, C), of a pair of groups (G, C) where C is a subgroup of G. Houghton uses rather different terminology from ours and his results are stated for topological groups. For simplicity, when quoting his results, we will rewrite them in our terminology and so as to apply to discrete groups only. Presumably, the main results of this paper can be generalised to topological groups in the same way that Abels [l] generalised Stallings' result [14]. My hope was to prove that e(G, C) 32 if and only if G splits over some finite extension of C. It turns out that this is false in general, but one can sometimes prove that e(G, C) 3 2 if and only if G has a subgroup of finite index which splits over a subgroup of G which is closely related to C. The first result on these lines was proved by Houghton [ll]. We state his result in Section 2. We will say that a group G with a subgroup C is C-residuaIZy fini& if given g in G-C, there is a subgroup G, of finite index in G such that G, contains C but not g. Our main result is the following. Theorem .I. If G and C are finitely generated groups and G is C-residually finite, then e(G, C) > 2 if and only if G has a subgroup G1 of finite index in G such that Cl contains C and G1 splits over C.

Statistical properties of subgroups of free groups

Random Structures & Algorithms, 2012

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: subgroups are generated (finite presentations are determined) by randomly chosen k-tuples of reduced words, whose maximal length is allowed to tend to infinity. In this paper we adopt a different, though equally natural point of view: we investigate the statistical properties of the same objects, but with respect to the so-called graph-based distribution, recently introduced by Bassino, Nicaud and Weil. Here, subgroups (and finite presentations) are determined by randomly chosen Stallings graphs whose number of vertices tends to infinity. Our results show that these two distributions behave quite differently from each other, shedding a new light on which properties of finitely generated subgroups can be considered frequent or rare. For example, we show that malnormal subgroups of a free group are negligible in the graphbased distribution, while they are exponentially generic in the word-based distribution. Quite surprisingly, a random finite presentation generically presents the trivial group in this new distribution, while in the classical one it is known to generically present an infinite hyperbolic group.