Boundaries and JSJ Decompositions of CAT(0)-Groups Article

First Online: 08 August 2009 Received: 01 January 2008 Revised: 15 September 2008 Accepted: 25 September 2008 DOI :
10.1007/s00039-009-0012-8

Cite this article as: Papasoglu, P. & Swenson, E. Geom. Funct. Anal. (2009) 19: 558. doi:10.1007/s00039-009-0012-8 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.

Keywords and phrases Boundary cut point JSJ decomposition closing lemma This work is co-funded by European Social Fund and National Resources (EPEAEK II) PYTHAGORAS.

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Authors and Affiliations 1. Mathematics Department University of Athens Athens Greece 2. Mathematics Department Brigham Young University Provo USA