Simple groups of finite Morley

The Algebraicity Conjecture treats model-theoretic foundations of algebraic group theory. It states that any simple group of finite Morley rank is an algebraic group over an algebraically closed field. In the mid-1990s a view was consolidated that this project falls into four cases of different flavour: even type, mixed type, odd type, and degenerate type. This book contains a proof of the conjecture in the first two cases, and much more besides: insight into the current state of the other cases (which are very much open), applications for example to permutation groups of finite Morley rank, and open questions. The book will be of interest to both model theorists and group theorists: techniques from the classification of finite simple groups (CFSG), and from other aspects of group theory (e.g., black box groups in computational group theory, and the theory of Tits buildings, especially of generalised polygons) play a major role. The techniques used are primarily group theoretic, but the history and motivation are more model theoretic. The origins of the conjecture, as with much modern model theory, lie in Morley’s Theorem: this states that if T is a complete theory in a countable first order language L (that is, T is a maximally consistent set of L-sentences) and T is κ-categorical for some uncountable cardinal κ, then T is κcategorical for all uncountable κ, that is, T is uncountably categorical. Here, the theory T is κ-categorical if all its models of size κ are isomorphic. Morley’s proof in [8] introduces the notion of Morley rank, an abstract dimension notion for definable sets (i.e., solution sets of formulas). In an instructive example, where T is the theory of an algebraically closed field K, definable sets are exactly the same as constructible sets (by quantifier elimination), and the Morley rank of a constructible set is just the algebraic-geometric dimension of its Zariski closure. Morley rank is ordinal valued: the Morley rank of a definable set X is at least α+1 if it is possible (at least after moving to an elementary extension) to partition X into infinitely many definable sets of rank at least α, and the definition at limit ordinals is the natural one. Morley showed that in an uncountably categorical theory, all definable sets have ordinal-valued Morley rank. In particular, such theories are stable (in fact, ω-stable). The rank was later shown by Baldwin to be finite. A new proof of Morley’s Theorem, yielding additional information, was given by Baldwin and Lachlan [1]. They showed that any uncountably categorical structure is “coordinatised” by a strongly minimal set, that is, a definable set all of whose definable subsets are finite or cofinite, uniformly in parameters. The form of coordinatisation was later refined by Zilber in his “Ladder theorem”; he also formulated conjectures on the structure of strongly minimal sets: obvious examples are pure sets (trivial geometry), vector spaces (locally modular geometry) and algebraically closed fields, and Zilber conjectured that any strongly minimal set with nonlocally modular geometry is very close to an algebraically closed field. This was proved false by Hrushovski [6]. With a new and delicate amalgamation technique