Finite Covers with Nite Kernels

We are concerned with the following problem. Suppose ? and are closed permutation groups on innnite sets C and W and : ? ?! is a non-split, continuous epimorphism with nite kernel. Describe (for xed) the possibilities for. Here, we consider the case where arises from a nite cover : C ?! W. We give reasonably general conditions on the permutation structure hW; i which allow us to prove that these covers arise in two possible ways. The rst way, reminiscent of covers of topological spaces, is as a covering of some-invariant digraph on W. The second construction is less easy to describe, but produces the most familiar of these types of covers: a vector space covering its projective space. Various natural model-theoretic questions can be cast into the following form: Given a structure, what additional structure can be put on one part of it without aaecting the structure on a diierent part? One such question which has received a certain amount of attention recently, largely as a result of work on totally categorical structures, is the problem of determining the nite covers of a given rst-order structure W (see, for example, 1, 2, 4, 5, 9, 10, 12, 13]). Loosely speaking, a nite cover C of W is obtained in two steps. First, one replaces each element of W by a copy of some nite structure, to obtain a free cover C 0. So W can be identiied with a subset of C eq 0. Then C is obtained from C 0 by adding extra relations on C 0 in such a way that no new deenable relations are induced on the copy of W. 1 The structures W considered in the papers cited above are all countable and @ 0-categorical. In these circumstances one tends to think (and work) in terms of permutation groups: if C is a nite cover of a countable, @ 0-categorical (or just !-saturated) structure W then there is a natural epimorphism : Aut(C) ?! Aut(W), between the automorphism groups. In this paper we study nite covers where the kernel of this epimorphism is nite, and (to avoid trivialites) the epimorphism is non-split. From a model-theoretic viewpoint, this special case of the problem plays an important role in the analysis of an arbitrary nite cover (see the main results, 2.1 and 4.7, of 5]). It is also the heart of the general problem in …