Let , for
, be the moduli
stack of –pointed,
stable complex curves of compact type. Various characterizations and properties are
obtained of both the topological and algebraic fundamental groups of the stack
instance we show that the topological fundamental groups are linear, extending to all
previous results of
Morita and Hain for
be the Teichmüller group associated with a compact Riemann surface of genus
ie the group of homotopy classes of diffeomorphisms of
which preserve the
orientation of and a given
order of its punctures. Let
be the normal subgroup of
generated by Dehn twists along separating simple closed curves (briefly s.c.c.) on
. The above theory yields
a characterization of
for all ,
improving Johnson’s classical results for closed and one-punctured surfaces in
[Topology 24 (1985) 113-126].
The Torelli group
is the kernel of the natural representation
. The abelianization
of the Torelli group
is determined for all
thus completing classical results of Johnson [Topology 24 (1985) 127-144] and Mess
[Topology 31 (1992) 775-790] for closed and one-punctured surfaces.
We also prove that a connected finite étale cover
, has a Deligne–Mumford
with finite fundamental group. This implies that, for
, any finite index
has vanishing first cohomology group, improving a result of Hain [Math. Sci. Res.
Inst. Publ. 28 (1995) 97-143].