An Improved Subgroup Theorem for HNN Groups with Some Applications

Karrass, A.; Pietrowski, Alfred; Solitar, D.

doi:10.4153/cjm-1974-021-1

Cited by 15 publications

(20 citation statements)

References 13 publications

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“…This is in turn a consequence of the fact that every finitely generated subgroup of such a group G is finitely presented (see [4] and groups of codimension < 2 is closed under forming amalgamated products and HNN extensions with free amalgamated and associated subgroups.…”

Section: Lemma a Finitely Generated Treed Hnn Group G (That Is Fundamentioning

confidence: 99%

A note on groups with separable finitely generated subgroups

Burns

Karrass

Solitar

1987

Bull. Austral. Math. Soc.

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An example is given of an infinite cyclic extension of a free group of finite rank in which not every finitely generated subgroup is finitely separable. This answers negatively the question of Peter Scott as to whether in all finitely generated 3-manifold groups the finitely generated subgroups are finitely separable. In the positive direction it is shown that in knot groups and one-relator groups with centre, the finitely generated normal subgroups are finitely separable.

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Section: Lemma a Finitely Generated Treed Hnn Group G (That Is Fundamentioning

confidence: 99%

A note on groups with separable finitely generated subgroups

Burns

Karrass

Solitar

1987

Bull. Austral. Math. Soc.

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“…The proof of the lemma is a simple application of the subgroup theorem for fundamental groups of graphs of groups. For a proof of the subgroup theorem, see [8] for the purely group-theoretic approach, see [2] for the approach via graphs of groups and trees, and see [21] for a topological approach. LEMMA 2.3.…”

Section: Finding An Irreducible 3-manifold-an Easy Special Casementioning

confidence: 99%

Fundamental Groups of Non-Compact 3-Manifolds

Scott

1977

Proceedings of the London Mathematical Society

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The aim of this paper is to consider certain results about compact 3-manifolds and their fundamental groups and to decide whether suitable generalizations hold in the non-compact case. We will say that a group G is a 3-manifold group if 0 is the fundamental group of some 3-manifold.Our starting point is the theorem of Kneser [9] which says that a compact 3-manifold is a finite connected sum of prime manifolds. In § 1, we discuss the idea of an infinite connected sum and we give an example of a non-compact 3-manifold which cannot be a finite or an infinite connected sum of prime manifolds. Our example is simply connected, so clearly this phenomenon is caused by geometric pathology and not by pathology of the fundamental group. Now suppose that M is a compact 3-manifold and that the fundamental group 0 of M is indecomposable (that is, O is not a non-trivial free product). Then Kneser's result tells us that there is a prime manifold N also with fundamental group G such that M is obtained from N by connected sum with 3-balls and homotopy 3-spheres. Our example mentioned above suggests that one cannot prove an analogous result when M is not compact. Instead we consider the weaker result which simply states that there exists a prime manifold iV with fundamental group G. Note that N is irreducible except when N is an # 2 -bundle over S 1 . In this exceptional case, we have that G is infinite cyclic and so G is the fundamental group of the solid torus S 1 x D 2 which is irreducible. Thus we see that there always exists an irreducible compact 3-manifold N with fundamental group G. We are unable to prove the obvious generalization of this result to the non-compact case, but in § 3 we prove the following theorem. THEOREM 3.1. If G is a 3-manifold group which is indecomposable and has no elements of order 2, then there exists a 3-manifold X with fundamental group G and 7T 2 {X) = 0.We remark that every embedded 2-sphere in X must bound a homotopy ball. Hence if the Poincare* conjecture is true, X must be irreducible. However, there is a special situation in which we are able to construct an

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“…Moreover, in [8] it is shown that use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700015445…”

Section: It Is Shown That I(a * B; F) = X(a) + X{b) -%{A N B) Where mentioning

confidence: 99%

Finite and infinite cyclic extensions of free groups

Karrass

Pietrowski

Solitar

1973

J. Aust. Math. Soc.

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130

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Using Stalling's characterization [11] of finitely generated (f. g.) groups with infinitely many ends, and subgroup theorems for generalized free products and HNN groups (see [9], [5], and [7]), we give (in Theorem 1) a n.a.s.c. for a f.g. group to be a finite extension of a free group. Specifically (using the terminology extension of and notation of [5]), a f.g. group G is a finite extension of a free group if and only if G is an HNN group where K is a tree product of a finite number of finite groups (the vertices of K), and each (associated) subgroup Li, Mi is a subgroup of a vertex of K.

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An Improved Subgroup Theorem for HNN Groups with Some Applications

Cited by 15 publications

References 13 publications

A note on groups with separable finitely generated subgroups

A note on groups with separable finitely generated subgroups

Fundamental Groups of Non-Compact 3-Manifolds

Finite and infinite cyclic extensions of free groups

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