The ambiguity index of an equipped finite group

Abstract

In the paper (Kulikov in Sb Math 204(2):237–263, 2013), 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 (Kunyavskiĭ in Cohomological and Geometric Approaches to Rationality Problems. Progress in Mathematics, vol 282, pp 209–217, 2010), see also (Bogomolov in Math USSR-Izv 30(3):455–485, 1988) 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.