From continua to ℝ–trees

Papasoglu, Panos; Swenson, Eric

doi:10.2140/agt.2006.6.1759

Cited by 15 publications

(49 citation statements)

References 7 publications

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“…Plan of the paper. In section 2 we give a summary of our earlier paper [21]. We explain how to produce trees from cut points and cut pairs.…”

Section: It Follows From This Theorem That If the Tits Diameter Of ∂Xmentioning

confidence: 99%

“…Bowditch ([8]) showed how to construct R-trees for boundaries of hyperbolic groups while Swenson ([28]) gave a more direct construction. In [21] we showed how to associate to a continuum Z an R-tree T encoding the cut points of Z. We recall here the results and terminology of [21].…”

Section: Trees and Continuamentioning

confidence: 99%

“…In [21] we showed how to associate to a continuum Z an R-tree T encoding the cut points of Z. We recall here the results and terminology of [21].…”

Section: Trees and Continuamentioning

confidence: 99%

“…We recall [21,Theorem 13] Theorem 12. The pretree P is preseparable, so the cut point tree T of Z is a real tree.…”

Section: Trees and Continuamentioning

confidence: 99%

“…We showed in [21] that if Z is a continuum without cut points, then we can associate to Z an R-tree T (called the JSJ tree of Z) encoding all pairs of points that separate Z.…”

Section: Introductionmentioning

confidence: 99%

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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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“…Plan of the paper. In section 2 we give a summary of our earlier paper [21]. We explain how to produce trees from cut points and cut pairs.…”

Section: It Follows From This Theorem That If the Tits Diameter Of ∂Xmentioning

confidence: 99%

Section: Trees and Continuamentioning

confidence: 99%

“…In [21] we showed how to associate to a continuum Z an R-tree T encoding the cut points of Z. We recall here the results and terminology of [21].…”

Section: Trees and Continuamentioning

confidence: 99%

“…We recall [21,Theorem 13] Theorem 12. The pretree P is preseparable, so the cut point tree T of Z is a real tree.…”

Section: Trees and Continuamentioning

confidence: 99%

“…We showed in [21] that if Z is a continuum without cut points, then we can associate to Z an R-tree T (called the JSJ tree of Z) encoding all pairs of points that separate Z.…”

Section: Introductionmentioning

confidence: 99%

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