G-Actions on Riemann Surfaces and the associated Group of Singular Orbit Data
Abstract
Let
G
be a finite group. To every smooth
G
-action on a compact, connected and oriented Riemann surface we can associate its data of singular orbits. The set of such data becomes an Abelian group
B
G
under the
G
-equivariant connected sum. The map which sends
G
to
B
G
is functorial and carries many features of the representation theory of finite groups. In this paper we will give a complete computation of the group
B
G
for any finite group
G
.
There is a surjection from the
G
-equivariant cobordism group of surface diffeomorphisms
Ω
G
to
B
G
. We will prove that the kernel of this surjection is isomorphic to
H
2
(G;Z)
. Thus
Ω
G
is an Abelian group extension of
B
G
by
H
2
(G;Z)
.
Finally we will prove that the group
B
G
contains only elements of order two if and only if every complex character of
G
has values in
R
. This property shows a strong relationship between the functor
B
and the representation theory of finite groups.