Degrees of isomorphism types and countably categorical groups (Q661283)

A countable structure with the universe \(\omega\) is said to be of (Turing) degree \(\mathbf{d}\) iff its atomic diagram has degree \(\mathbf{d}\). By the spectrum \(\mathrm{Spec}(\mathcal{M})\) of a structure \(\mathcal{M}\) we mean the set of degrees of structures isomorphic to \(\mathcal{M}\). If \(\mathrm{Spec}(\mathcal{M})\) has a least element \(\mathbf{d}\) then \(\mathcal{M}\) is said to have degree \(\mathbf{d}\); otherwise, we say that \(\mathcal{M}\) has no degree. The main result of the paper under review is following: Theorem. (i) For every Turing degree \(\mathbf{d}\) there is a 2-step nilpotent group \(\mathfrak{G}\) of exponent four so that \(\mathfrak{G}\) has a countably categorical theory submodel complete (i.e., with quantifier elimination) theory and has degree \(\mathbf{d}\). (ii) There is a 2-step nilpotent group \(\mathfrak{G}\) of exponent four so that \(\mathfrak{G}\) has a countably categorical theory submodel complete theory and has no degree. The author uses the Fraïssé method for constructing both series of examples.