On Torsion-Free Groups with Infinitely Many Ends

Stallings, John R.

doi:10.2307/1970577

Cited by 382 publications

(251 citation statements)

References 12 publications

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“…In [18], [19] Stallings showed that a finitely generated group G splits over a finite subgroup if and only if G has more than one end.…”

Section: Introductionmentioning

confidence: 99%

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A Geometric Proof of Stallings' Theorem on Groups with More than One End

Niblo

2004

Geometriae Dedicata

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Abstract. Stallings showed that a finitely generated group which has more than one end splits as an amalgamated free product or an HNN extension over a finite subgroup. Dunwoody gave a new geometric proof of the theorem for the class of almost finitely presented groups. Here we adapt the method to the class of finitely generated groups using Sageev's generalisation of Bass Serre theory concerning group pairs with more than one end.

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“…In [18], [19] Stallings showed that a finitely generated group G splits over a finite subgroup if and only if G has more than one end.…”

Section: Introductionmentioning

confidence: 99%

“…Stallings remarked in [18] that he was led to the proof by consideration of Papakyriokopoulos's sphere theorem for 3-manifolds which may be understood in terms of minimal surface theory.…”

Section: Introductionmentioning

confidence: 99%

A Geometric Proof of Stallings' Theorem on Groups with More than One End

Niblo

2004

Geometriae Dedicata

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“…The following was proven by Stallings [10] for torsion-free groups, his proof was extended by Bergman [2] to groups with torsion:…”

Section: Proof Of Stallings' Theoremmentioning

confidence: 94%

Energy of harmonic functions and Gromov's proof of Stallings' theorem

Kapovich

2014

Georgian Mathematical Journal

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“…Hence, cd G = 1, but then by [Stallings 1968] and [Swan 1969], G must itself be free, a contradiction.…”

Section: Final Remarksmentioning

confidence: 98%

Parasurface groups

Bou-Rabee¹

2010

Pacific J. Math.

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A residually nilpotent group is k-parafree if all of its lower central series quotients match those of a free group of rank k. Magnus proved that kparafree groups of rank k are themselves free. We mimic this theory with surface groups playing the role of free groups. Our main result shows that the analog of Magnus' theorem is false in this setting. IntroductionThis article is motivated by three stories. The first story concerns a theorem of Wilhelm Magnus. Recall that the lower central series of a group G is defined to bewhere [A, B] denotes the group generated by commutators of elements of A with elements of B. The rank of G is the size of a minimal generating set of G. In [1939], Magnus gave a beautiful characterization of free groups in terms of their lower central series.Theorem (Magnus' theorem on parafree groups). Let F k be a nonabelian free group of rank k and G a group of rank k.Following this result, Hanna Neumann inquired whether it was possible for two residually nilpotent groups G and G to have G/γ i (G) ∼ = G /γ i (G ) for all i without having G ∼ = G ; see [Liriano 2007]. Gilbert Baumslag [1967] gave a positive answer to this question by constructing what are now known as parafree groups that are not themselves free. A group G is parafree if(1) G is residually nilpotent, and (2) there exists a finitely generated free group F such that G/γ i (G) ∼ = F/γ i (F) for all i.MSC2000: 20F22.

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On Torsion-Free Groups with Infinitely Many Ends

Cited by 382 publications

References 12 publications

A Geometric Proof of Stallings' Theorem on Groups with More than One End

A Geometric Proof of Stallings' Theorem on Groups with More than One End

Energy of harmonic functions and Gromov's proof of Stallings' theorem

Parasurface groups

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