Hirotaka Kikyo (Kobe University)

On some generic structures

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This is a joint work with Yutaka Kuga, a student of mine.

The talk is about the generic structures produced by Hrushovski's predimension construction with a control function.

A predimension of a graph is the number of vertices minus the number of edges multiplied by some weight. With a predimension and some control function $f$, a class of finite graphs $\mathrm{K}_f$ is defined. Suppose $f$ is unbounded, $\mathrm{K}_f$ has the free amalgamation property and one point substructures are always closed in a sense defined by the predimension. Let $M$ be the generic structure of $\mathrm{K}_f$. Our result is that $\operatorname{Th}(M)$ is model complete if the weight of the predimension is a rational number. In the case that it is an irrational number, $\operatorname{Th}(M)$ is also model complete if $f$ satisfies some mild assumptions satisfied by all known examples of such $f$.

Using the techniques used in the proof of model completeness of $\operatorname{Th}(M)$, we can also show that $M$ is monodimensional in the case that the weight of the predimension is rational and $f$ is the function defined by Hrushovski in his original paper. Hence, the automorphism group of $M$ is a simple group (has no non-trivial normal groups) by a theorem of Evans, Ghadernezhad, and Tent. The same result is valid for most examples of such $f$.