  Let S be a surface and let Mod(S,K) be the mapping class group of S permuting
a Cantor subset K of S. We prove two structure theorems for normal subgroups of
Mod(S,K).
  (Purity:) if S has finite type, every normal subgroup of Mod(S,K) either
contains the kernel of the forgetful map to the mapping class group of S, or it
is `pure', i.e. it fixes the Cantor set pointwise.
  (Inertia:) for any n element subset Q of the Cantor set, there is a forgetful
map from the pure subgroup PMod(S,K) of Mod(S,K) to the mapping class group of
(S,Q) fixing Q pointwise. If N is a normal subgroup of Mod(S,K) contained in
PMod(S,K), its image N_Q is likewise normal. We characterize exactly which
finite-type normal subgroups N_Q arise this way.
  Several applications and numerous examples are also given.
