Strong orbit equivalence in Cantor dynamics and simple locally finite groups

In this article, we give a dynamical and elementary proof of a result of Giordano, Putnam and Skau which establishes a necessary and sufficient condition for two minimal homeomorphisms of a Cantor space to be strong orbit equivalent. Our argument is based on a detailed study of some countable locally finite groups attached to minimal homeomorphisms. This approach also enables us to prove that the Borel complexity of the isomorphism relation on simple locally finite groups is a universal relation arising from a Borel S∞-action.