G-Actions on Riemann Surfaces and the associated Group of Singular Orbit Data

TL;DR

This paper computes the group of singular orbit data for smooth finite group actions on Riemann surfaces, revealing its structure, relation to cobordism, and connections to the representation theory of finite groups.

Contribution

It provides a complete computation of the group of singular orbit data for any finite group and establishes its relation to equivariant cobordism and representation theory.

Findings

The group $B_G$ is explicitly computed for all finite groups.

The kernel of the surjection from $ ext{Omega}_G$ to $B_G$ is isomorphic to $H_2(G;Z)$.

$B_G$ contains only elements of order two if and only if all complex characters of $G$ have values in $R$.

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 $\Omega_G$ to $B_G$. We will prove that the kernel of this surjection is isomorphic to $H_2(G;Z)$. Thus $\Omega_G$ 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.