Finite covers appeared in model theory in attempts to classify strongly minimal structures (see [D. M. Evans, D. Macpherson and A. A. Ivanov, in: D. M. Evans (ed.), Model theory of groups and automorphism groups. Lectures held at the RESMOD summer school, Blaubeuren, Germany, July 31–August 5, 1995. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 244, 1–72 (1997; Zbl 0888.03020)] for a general theory of finite covers). Most of the known results concerned the \(\omega\)-categorical case. The author considers finite covers of the Cayley graphs of infinite finitely generated groups (such graphs are natural examples of strongly minimal non-\(\omega\)-categorical structures in a finite language), and is interested in the following questions. When is a finite cover finitely axiomatizable over the base? When does a finite cover have the finite model property?
G. Ahlbrandt and M. Ziegler [Ann. Pure Appl. Logic 30, No. 1, 63–82 (1986; Zbl 0592.03018)] introduced the notion of a nice enumeration as a tool in the study of finite covers of strictly minimal \(\omega\)-categorical structures. An enumeration \(w_0, w_1,\dots\) of a structure \(W\) is called nice if any set \(P\) of nice pairs has a finite subset \(P'\) such that for any \((w,S)\in P\) there are \((w',S')\in P'\) and \(\alpha\in \text{Aut}(W)\) with \(\alpha(w')=w\) and \(\alpha(S')\subseteq S\), where the nice pairs are defined to be the \(\text{Aut}(W)\)-conjugates of pairs of the form \((w_n,\{w_0,\dots,w_{n-1}\})\). Similarly, one can define nice enumerations for any permutation structure \((G,W)\).
Let \(W\) be the Cayley graph of an infinite finitely generated group \(G\), and \(M\) a finite cover of \(W\). The author proves that if \(W\) has a nice enumeration then \(\text{Th}(M)\) is finitely axiomatizable over \(\text{Th}(W)\). He shows that \(W\) has a nice enumeration in case of virtually abelian \(G\), and conjectures that if the Cayley graph of any subgroup of finite index in \(G\) has a “natural” nice enumeration then \(G\) is virtually abelian. The conjecture is verified for solvable \(G\). In case of virtually abelian \(G\), it is proven that \(\text{Th}(M)\) is not finitely axiomatizable, but it is open whether \(\text{Th}(M)\) has the finite model property; the latter is verified for \(G=\mathbb{Z}^2\). Also, it is shown that \(\text{Th}(M)\) has the finite model property if \(G\) is virtually nilpotent and the kernel of the cover is finite.
A number of results on non-existence of nice enumerations of non-disintegrated strongly minimal structures are proven: there is no nice enumeration of (1) a locally modular strongly minimal non-\(\omega\)-categorical group (with extra structure), (2) a countable saturated non-locally-modular strongly minimal structure, (3) the permutation structure \((\text{AGL}_n(D), D^n)\), for any infinite field \(D\).