Model Theory of Finite Difference Fields and Simple Groups

Asymptotic classes are classes of finite structures which have uniformly definable estimates for the cardinalities of their first-order definable sets akin to those in finite fields given by the Lang-Weil estimates. The goal of the thesis is to prove that the finite simple groups of a fixed Lie type and Lie rank form asymptotic classes. This requires the following: 1. The introduction describes the background. 2. Chapter 4 shows a general method of generating one asymptotic class of structures from another through the notion of bi-interpretability. Specifically, the notions we introduce are those of uniform parameter bi-interpretability and strong uniform parameter bi-interpretability. We prove that being an asymptotic class is preserved under strong uniform parameter bi-interpretability. 3. Chapter 5 shows that classes of finite simple groups of a fixed Lie type and Lie rank are strongly uniformly parameter bi-interpretable with specific classes of finite fields or finite difference fields. This reduces our task to demonstrating that certain classes of finite difference fields form asymptotic classes. 4. Chapter 2 yields a definability of measure result for the finite σ-degree sets in the theory ACFA. The principal result of the chapter is Theorem 2.1.1, and it is published in work with Ivan Tomasic, in [25]. In a similar vein, Chapter 3 develops the asymptotic theory of finite fields equipped with fractional powers of the Frobenius. Equipped with this almost theory, we demonstrate the existence of many asymptotic classes of finite difference fields. In particular, we demonstrate that the classes of finite difference fields found in Chapter 5 to be uniformly parameter bi-interpretable with certain classes of finite simple groups do form asymptotic classes. Combining our results we achieve the goal of the thesis.