Geometric and Functional Analysis

, Volume 19, Issue 2, pp 558–590 | Cite as

Boundaries and JSJ Decompositions of CAT(0)-Groups



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.

Keywords and phrases

Boundary cut point JSJ decomposition closing lemma 

2000 Mathematics Subject Classification

20F67 20E06 20E34 57M07 


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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of AthensAthensGreece
  2. 2.Mathematics DepartmentBrigham Young UniversityProvoUSA

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