The algebraic torus theorem

Dunwoody, M. J.; Swenson, Eric

doi:10.1007/s002220000063

Cited by 46 publications

(81 citation statements)

References 23 publications

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“…The technique described in this paper can also be applied to obtain a new proof of the following generalisation of Stallings' theorem which was known to Dunwoody and Roller [6]; it plays a key role in the proof of the algebraic torus theorem [7]. The result was (re)proved in [10] by appealing to Dunwoody's result on cuts in graphs [4].…”

Section: Theorem (Dunwoody)mentioning

confidence: 99%

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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Section: Theorem (Dunwoody)mentioning

confidence: 99%

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

Niblo

2004

Geometriae Dedicata

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“…This is immediate from Theorem 3.11 and Theorem 4.2, which says one can read any almost invariant set over a VPC n subgroup in a JSJ tree T , and which follows from [Dunwoody and Swenson 2000] and [Scott and Swarup 2003, Theorem 8.2].…”

Section: The Regular Neighborhood Of Scott and Swarupmentioning

confidence: 96%

“…Theorem 4.2 is essentially another take on the proof of Scott and Swarup's [2003, Theorem 8.2], and makes a crucial use of algebraic torus theorems of [Dunwoody and Swenson 2000;Dunwoody and Roller 1993]. We give a proof in Section 4.…”

Section: Introductionmentioning

confidence: 99%

Scott and Swarup’s regular neighborhood as a tree of cylinders

Guirardel¹,

Levitt²

2010

Pacific J. Math.

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“…By theorem 49 there is a hyperbolic element g in V with {g ∞ , g −∞ } = {a, b}. By the algebraic torus theorem [11] to show that G splits over a 2-ended group it suffices to show that X/ < g > has more than one end. LetX = X ∪∂X.…”

Section: 3mentioning

confidence: 99%

Boundaries and JSJ Decompositions of CAT(0)-Groups

Papasoglu¹,

Swenson

2009

Geom. Funct. Anal.

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Abstract. Let G be a one-ended group acting discretely and co-compactly on a CAT(0) space X. We show that ∂X has no cut points and that one can detect splittings of G over two-ended groups and recover its JSJ decomposition from ∂X.We show that any discrete action of a group G on a CAT(0) space X satisfies a convergence type property. This is used in the proof of the results above but it is also of independent interest. In particular, if G acts co-compactly on X, then one obtains as a Corollary that if the Tits diameter of ∂X is bigger than 3π 2 then it is infinite and G contains a free subgroup of rank 2.

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The algebraic torus theorem

Cited by 46 publications

References 23 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

Scott and Swarup’s regular neighborhood as a tree of cylinders

Boundaries and JSJ Decompositions of CAT(0)-Groups

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