## The ambiguity index of an equipped finite group.(English)Zbl 1343.20017

Summary: In the paper [V. S. Kulikov, Sb. Math. 204, No. 2, 237-263 (2013); translation from Mat. Sb. 204, No. 2, 87-116 (2013; Zbl 1290.14020)], 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 [B. Kunyavskiĭ, Prog. Math. 282, 209-217 (2010; Zbl 1204.14006)], see also [F. A. Bogomolov, Math. USSR, Izv. 30, No. 3, 455-485 (1988); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 51, No. 3, 485-516 (1987; Zbl 0679.14025)] and hence can be easily computed for many pairs $$(G,O)$$. In particular, the ambiguity indices are completely counted in the cases when $$G$$ are the symmetric or alternating groups.

### MSC:

 20C25 Projective representations and multipliers 14H30 Coverings of curves, fundamental group

### Citations:

Zbl 1290.14020; Zbl 1204.14006; Zbl 0679.14025
Full Text:

### References:

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