Boundaries and JSJ Decompositions of CAT(0)-Groups
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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 phrasesBoundary cut point JSJ decomposition closing lemma
2000 Mathematics Subject Classification20F67 20E06 20E34 57M07
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