  In this article, we study the smooth mapping class group of a surface S
relative to a given Cantor set, that is the group of isotopy classes of
orientation-preserving smooth diffeomorphisms of S which preserve this Cantor
set. When the Cantor set is the standard ternary Cantor set, we prove that the
subgroup consisting of diffeomorphisms which are isotopic to the identity on S
does not contain any distorted elements. Moreover, we prove a weak Tits
alternative for these groups.
