Bogomolov, Fedor A.; Kulikov, Viktor S. The ambiguity index of an equipped finite group. (English) Zbl 1343.20017 Eur. J. Math. 1, No. 2, 260-278 (2015). 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. Cited in 2 Documents MSC: 20C25 Projective representations and multipliers 14H30 Coverings of curves, fundamental group Keywords:finite groups; ambiguity index; equipped groups; \(C\)-groups; Bogomolov multipliers; Hurwitz spaces Citations:Zbl 1290.14020; Zbl 1204.14006; Zbl 0679.14025 PDFBibTeX XMLCite \textit{F. A. Bogomolov} and \textit{V. S. Kulikov}, Eur. J. Math. 1, No. 2, 260--278 (2015; Zbl 1343.20017) Full Text: DOI arXiv References: [1] Bogomolov, F.A.: The Brauer group of quotient spaces of linear representations. Math. USSR-Izv. 30(3), 455-485 (1988) · Zbl 0679.14025 [2] Bogomolov, F., Maciel, J., Petrov, T.: Unramified Brauer groups of finite simple groups of Lie type \[A_\ell\] Aℓ. Am. J. Math. 126(4), 935-949 (2004) · Zbl 1058.14031 [3] Clebsch, A.: Zur Theorie der Riemann’schen Fläche. Math. Ann. 6(2), 216-230 (1873) · JFM 05.0227.01 [4] Fulton, W.: Hurwitz schemes and irreducibility of moduli of algebraic curves. Ann. Math. 90(3), 542-575 (1969) · Zbl 0194.21901 [5] Hoffman, P.N., Humphreys, J.F.: Projective Representations of the Symmetric Groups. In: Oxford Mathematical Monographs. Clarendon Press/Oxford University Press, New York (1992) · Zbl 0777.20005 [6] Hurwitz, A.: Ueber Riemann’sche Flächen mit gegebenen Verzweigungspunkten. Math. Ann. 39(1), 1-60 (1891) · JFM 23.0429.01 [7] Jezernik, U., Moravec, P.: Universal commutator relations, Bogomolov multipliers, and commuting probability. arXiv:1307.6533 (2013) · Zbl 1314.14024 [8] Kang, M., Kunyavskiĭ, B.: The Bogomolov multiplier of rigid finite groups. Arch. Math. (Basel) 102(3), 209-218 (2014) · Zbl 1328.14076 [9] Kulikov, Vik.S.: Hurwitz curves. Rus. Math. Surv. 62(6), 1043-1119 (2007) · Zbl 1141.14311 [10] Kulikov, Vik.S.: Factorization semigroups and irreducible components of the Hurwitz space. II. Izv. Math. 76(2), 356-364 (2012) · Zbl 1251.14018 [11] Kulikov, Vik.S.: Factorizations in finite groups. Sb. Math. 204(2), 237-263 (2013) · Zbl 1290.14020 [12] Kulikov, Vik.S., Kharlamov, V.M.: Covering semigroups. Izv. Math. 77(3), 594-626 (2013) · Zbl 1286.14045 [13] Kunyavskiĭ, B.: The Bogomolov multiplier of finite simple groups. In: Cohomological and Geometric Approaches to Rationality Problems. Progress in Mathematics, vol. 282, pp. 209-217. Birkhäuser, Boston (2010) · Zbl 1204.14006 [14] Saltman, D.J.: Noether’s problem over an algebraically closed field. Invent. Math. 77(1), 71-84 (1984) · Zbl 0546.14014 [15] Spanier, E.H.: Algebraic Topology. Springer, New York (1981) · Zbl 0145.43303 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.