Metric Sparsitication Under Finite Decomposition Complexity

The notions of metric sparsification property and finite decomposition complexity are recently introduced in metric geometry to study the coarse Novikov conjecture and the stable Borel conjecture.In this paper we prove that a metric space X has finite decomposition complexity with respect to metric sparsification property if and only if X itself has metric sparsification property.As a consequence,we obtain an alternative proof to a very recent result by E.Guentner,R.Tessera and G.Yu that all countable linear groups have the metric sparsification property and hence the operator norm localization property.