A dynamical proof of Matui's absorption theorem

TL;DR

This paper provides a new, elementary dynamical proof of Matui's absorption theorem and applies it to fix a gap in a classification proof of orbit equivalence for minimal ample groups.

Contribution

It introduces a more accessible proof of Matui's absorption theorem and uses it to correct a previous classification proof in dynamical systems.

Findings

A new dynamical proof of Matui's absorption theorem

Correction of the orbit equivalence classification proof

Enhanced understanding of minimal ample group dynamics

Abstract

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).