A dynamical proof of Matui's absorption theorem

We give a dynamical, relatively elementary proof of an “absorption theorem” which is closely related to a well-known result due to Matui. The construction is in the spirit of an earlier joint work of the author and S. Robert. In an appendix we explain how to use this result to correct the dynamical proof given by Melleray–Robert of a classification theorem for orbit equivalence of minimal ample groups due to Giordano, Putnam and Skau (the original argument had a gap).