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Published online by Cambridge University Press: 27 August 2018
The aim of this chapter is to provide some new tools to aid the study of decomposition complexity, a notion introduced by Guentner, Tessera and Yu. In this chapter, three equivalent definitions for decomposition complexity are established. We prove that metric spaces with finite hyperbolic dimension have finite (weak) decomposition complexity, and we prove that the collection of metric families that are coarsely embeddable into Hilbert space is closed under decomposition. A method for showing that certain metric spaces do not have finite decomposition complexity is also discussed.
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