Finite covers with finite kernels
The author considers certain infinite imprimitive automorphism groups, motivated by model-theoretic questions on classes of \(\aleph_0\)-categorical structures. The basic notion is that of a permutation structure, that is a set \(X\) together with a closed subgroup of \(\text{Sym}(X)\) (denoted \(\text{Aut}(X)\)). If \(C\), \(W\) are permutation structures, then a finite cover is a surjection \(\pi:C\rightarrow W\) such that the \(\{\pi^{-1}(w)\}\) (where \(w \in W\)) are finite and form an \(\text{Aut}(C)\)-invariant partition of \(C\), and the induced map \(\rho: \text{Aut}(C) \rightarrow \text{Aut}(W)\) given by \(\rho g(w)=\pi(g\pi^{-1}(w))\) has image \(\text{Aut}(W)\). The kernel of the cover is \(\text{ker}(\mu)\). The emphasis is on the case when \(W\) is \(\aleph_0\)-categorical (so \(\text{Aut}(W)\) has finitely many orbits on \(k\)-sets for all finite \(k\)) and \(\text{ker}(\mu)\) is finite. The author examines possible kernels, and possible covers with given kernel, under certain hypotheses. In particular, he conjectures that under natural hypotheses (on \(W\) and the class of covers), for a given permutation structure \(W\) there is a natural number \(r\) such that the kernel of any finite cover is at most \(r\)-generated. This has since been verified, by the author [J. Algebra 193, 214-238 (1997; Zbl 0884.03041)] and also by J. Koshan and by E. Hrushovski. The author introduces natural conditions -- untwisted, locally trivial, locally transitive (sometimes relativised to an orbit) -- and gives reduction lemmas to these situations. He develops a theory of finite coverings of digraphs, and uses this, under certain hypotheses, to define a universal covering of a structure, and also to define an analogue of the first homology group, which has all sufficiently nice kernels as quotients. This theory works well if there is a strong type, a privileged 1-type which exists in many \(\omega\)-categorical structures. In the final section, the author gives a reduction to the locally transitive situation. (The natural example is an \(\aleph_0\)-dimensional vector space over a finite field, covering projective space.) Assuming the existence of a strong type, sufficiently nice finite covers are parametrised.
scientific article; zbMATH DE number 1109921