Computation of first cohomology groups of finite covers
The paper contains important structural results on finite covers of permutation structures (in the context of infinite permutation groups). These arise naturally in model theory, for example in the analysis of totally categorical (or, much more generally, smoothly approximable) structures. The problem is, given a large base structure \(W\), to determine all finite covers of \(W\) (finite means that the fibres are finite.) The proofs are cohomological, typically using Shapiro's Lemma. Theorem 1.5 (also proved by Hrushovski) states that if \(W\) is an oligomorphic \(G\)-finite permutation structure, then there is a (computable) bound \(r\) such that if \(\pi:C \rightarrow W\) is a minimal finite cover with finite kernel \(K\), then \(K\) has a generating set of size at most \(r\) (minimal here means that no proper expansion of \(C\) induces the same structure on \(W\).) In Corollary 5.4, using Sylow theory for profinite groups, it is shown that for a permutation structure \(W\) on which \(\Aut(W)\) has finitely many orbits, the kernel of any minimal finite cover is nilpotent. Then in Theorem 6.1, sufficient conditions are given for a minimal finite cover of \(W\) to have finite kernel. Finally, in Section 7, the author treats generalisations of Ziegler's Finiteness Conjecture. He shows that, under a certain combinatorial hypothesis on \(W\), each cover of \(W\) is determined up to finitely many possibilities by its kernel. The number of such covers is bounded in terms of the order of a certain cohomology group, which is shown to be finite (but the method does not give explicit bounds).
scientific article; zbMATH DE number 1051519