, Volume 19, Issue 2, pp 558-590
Date: 08 Aug 2009

Boundaries and JSJ Decompositions of CAT(0)-Groups

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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.

This work is co-funded by European Social Fund and National Resources (EPEAEK II) PYTHAGORAS.