The ambiguity index of an equipped finite group

TL;DR

This paper links the ambiguity index of equipped finite groups to a generalized Bogomolov multiplier, enabling easier computation and deeper understanding of the structure of Hurwitz spaces and Galois coverings.

Contribution

It establishes a connection between the ambiguity index and a generalized Bogomolov multiplier, facilitating computation for various equipped finite groups.

Findings

Ambiguity index equals the size of a generalized Bogomolov multiplier.

The ambiguity index can be computed explicitly for many pairs (G,O).

Provides a new algebraic tool for studying Hurwitz spaces and Galois coverings.

Abstract

In \cite{Ku0}, the ambiguity index $a_{(G,O)}$ was introduced for each equipped finite group $(G,O)$. It is equal to the number of connected components of a Hurwitz space parametrizing coverings of a projective line with Galois group $G$ assuming that all local monodromies belong to conjugacy classes $O$ in $G$ and the number of branch points is greater than some constant. We prove in this article that the ambiguity index can be identified with the size of a generalization of so called Bogomolov multiplier (\cite{Kun1}, see also \cite{BO87}) and hence can be easily computed for many pairs $(G,O)$.