Permanence of Metric Sparsification Property 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, it is proved 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, the authors obtain an alternative proof of a very recent result by Guentner, Tessera and Yu that all countable linear groups have the metric sparsification property and hence the operator norm localization property.