Bounds in the theory of finite covers (Q1604371)

Let \(\Gamma_0\) be the unrestricted wreath product \(F\text{ Wr}_W\text{Sym}(\Omega)\), where \(F\) is a finite nilpotent group, \(\Omega\) is an infinite set, and \(W\) is the collection of \(n\)-tuples of distinct elements from \(\Omega\) (with \(\text{Sym}(\Omega)\) acting naturally). The authors give an upper bound (in terms of \(n\) and the composition length of \(F\)) on the number of conjugacy classes of closed subgroups of \(\Gamma_0\) which project onto \(\text{Sym}(\Omega)\). The methods include an extension to infinite symmetric groups of results on representations of finite symmetric groups, some group cohomology, and a result of Krajíček on the weight of generating sets for submodules of tabloid modules for the infinite degree symmetric groups. Part of the original motivation for this was model-theoretic. Any disintegrated totally categorical structure is bi-interpretable with a finite cover of a Grassmannian of a pure set. It is known that if the arity and the fibre group of such a structure is fixed, then the number of possible covers is finite. The present paper gives bounds in certain cases.