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