Subgroups of HNN groups
01 Jun 1974-Journal of The Australian Mathematical Society (Cambridge University Press)-Vol. 17, Iss: 4, pp 394-405
TL;DR: In this article, a more precise form of Theorem 1 of [2] was given, which gives a structure theorem for subgroups of HNN groups; this was later extended in [3] and [4].
Abstract: The purpose of this paper is to give a more precise form of Theorem 1 of [2], which gives a structure theorem for subgroups of HNN groups; we prove the following.Let H be a subgroup of the HNN group . Then H is an HNN group whose base is a tree product of groups H ∪ wAw-1 where w runs over a set of double coset representatives of (H,A); the amalgamated and associated subgroups are all of the form H ∊ vUiv-l for some v. We can be more precise about which subgroups occur and about the tree product. We will also obtain stronger forms of other results in [1] and [2].
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Book•
01 Jan 1975
TL;DR: There is a sympathy of ideas among the fields of knot theory, infinite discrete group theory, and the topology of 3-manifolds as mentioned in this paper, and several papers have been contributed by Goldsmith, Levine, Lomonaco, Perko, Trotter, and Whitten.
Abstract: There is a sympathy of ideas among the fields of knot theory, infinite discrete group theory, and the topology of 3-manifolds. This book contains fifteen papers in which new results are proved in all three of these fields. These papers are dedicated to the memory of Ralph H. Fox, one of the world's leading topologists, by colleagues, former students, and friends. In knot theory, papers have been contributed by Goldsmith, Levine, Lomonaco, Perko, Trotter, and Whitten. Of these several are devoted to the study of branched covering spaces over knots and links, while others utilize the braid groups of Artin. Cossey and Smythe, Stallings, and Strasser address themselves to group theory. In his contribution Stallings describes the calculation of the groups In/In+1 where I is the augmentation ideal in a group ring RG. As a consequence, one has for each k an example of a k-generator l-relator group with no free homomorphs. In the third part, papers by Birman, Cappell, Milnor, Montesinos, Papakyriakopoulos, and Shalen comprise the treatment of 3-manifolds. Milnor gives, besides important new results, an exposition of certain aspects of our current knowledge regarding the 3- dimensional Brieskorn manifolds.
41 citations
01 May 2008
TL;DR: In this article, a family of groups and a free group are considered, and the free product of the family is the product of all the groups in the family and the group.
Abstract: Let (Gi j i 2 I) be a family of groups, let F be a free group, and let G = F ⁄ ⁄ i2I Gi, the free product of F and all the Gi
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TL;DR: In this article, it was shown that fully residually free groups have the Howson property that the intersection of any two finitely generated subgroups in such a group is again finitely created.
Abstract: We prove that fully residually free groups have the Howson property, that is the intersection of any two finitely generated subgroups in such a group is again finitely generated. We also establish some commensurability properties for finitely generated fully residually free groups which are similar to those of free groups. Finally we prove that for a finitely generated fully residually free group the membership problem is solvable with respect to any finitely generated subgroup.
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TL;DR: In this article, it was shown that all subgroups H of a free product G of two groups A, B with an amalgamated subgroup V are obtained by two constructions from the intersection of H and certain conjugates of A, b, and U. The constructions are those of a tree product, a special kind of generalized free product, and of a Higman-NeumannNeumann group.
Abstract: We prove that all subgroups H of a free product G of two groups A, B with an amalgamated subgroup V are obtained by two constructions from the intersection of H and certain conjugates of A, B, and U. The constructions are those of a tree product, a special kind of generalized free product, and of a Higman-NeumannNeumann group. The particular conjugates of A, B, and U involved are given by double coset representatives in a compatible regular extended Schreier system for G modulo H. The structure of subgroups indecomposable with respect to amalgamated product, and of subgroups satisfying a nontrivial law is specified. Let A and B have the property P and U have the property Q. Then it is proved that G has the property P in the following cases: P means every f.g. (finitely generated) subgroup is finitely presented, and Q means every subgroup is f.g.; P means the intersection of two f.g. subgroups is f.g., and Q means finite; P means locally indicable, and Q means cyclic. It is also proved that if A' is a f.g. normal subgroup of G not contained in U, then NU has finite index in G.
187 citations
TL;DR: In this article, the authors use the results in [6] to obtain a structure theorem for the subgroups of an HNN group and give several applications, e.g., the notion of a tree product as defined in [7] will also be needed.
Abstract: HNN groups have appeared in several papers, e.g., [3; 4; 5; 6; 8]. In this paper we use the results in [6] to obtain a structure theorem for the subgroups of an HNN group and give several applications. We shall use the terminology and notation of [6]. In particular, if K is a group and {φ i } is a collection of isomorphisms of subgroups {L i} into K, then we call the group 1 the HNN group with base K, associated subgroups { Li,φi (Li )} and free part the group generated by t1, t2, …. (We usually denote φi (Li ) by Mi or L –i.) The notion of a tree product as defined in [6] will also be needed.
92 citations