In this appendix we consider smooth volume preserving action of a countable group on a compact manifold. It is proved that if the derivative cocycle of the action has integrable norm then the action has discrete spectrum. In particular if the group action preserves a measurable Riemannian structure, which has square integrable distortion with respect to the smooth structure, the action has discrete spectrum. This complements the constructions in the main paper. The short proof is based on the notion of entropy of a sequence of m.p. transformations, and an estimate of such entropy by the logarithm of the $L^1$-norm of the derivative cocycle.