Diffeomorphism groups of tame Cantor sets and Thompson-like groups

The group of C-1-diffeomorphisms of any sparse Cantor subset of a manifold is countable and discrete (possibly trivial). Thompson's groups come out of this construction when we consider central ternary Cantor subsets of an interval. Brin's higher-dimensional generalizations nV of Thompson's group V arise when we consider products of central ternary Cantor sets. We derive that the C-2-smooth mapping class group of a sparse Cantor sphere pair is a discrete countable group and produce this way versions of the braided Thompson groups.