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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
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