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## The ambiguity index of an equipped finite group

European Journal of Mathematics. 2015. Vol. 1. No. 4. P. 260-278.
F. A. Bogomolov, Vik. S. Kulikov.

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)\$.