  The group of $\mathcal C^1$-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 $\mathcal C^2$-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.
