On the automorphism groups of finite covers

Abstract

We are concerned with identifying by how much a finite cover of an ℵ 0-categorical structure differs from a sequence of free covers. The main results show that (in the best circumstances) this is measured by automorphism groups which are nilpotent-by-abelian. In the language of covers, these results say that every finite (regular) cover can be decomposed naturally into linked, superlinked and free covers. The superlinked covers arise from covers over a different base, and to describe this properly we introduce the notion of a quasi-cover. These results generalise results of the second author obtained in the case where the base of the cover is a grassmannian of a disintegrated set. They also give a complete proof of a statement of the second author extending this case to the case of a grassmannian of a modular set. To do this, we need to analyse the possible superlinked covers of such a set. We also give a combinatorial condition on the base of a cover which guarantees various chain conditions on finite covers over this base, and introduce a pregeometry which is useful in the analysis of finite covers with simple fibre groups.