Tame Groups, Universal Graphs, Automorphism Towers, and Cofinalities of Infinite Groups

Cherlin proposes to collaborate with a team of eight researchers on the classification of connected simple tame groups of finite Morley rank, using methods suggested by experience with finite groups: it is anticipated that all such groups are algebraic. In addition Cherlin, working with Shi, seeks to introduce model theoretic methods into the study of universal graphs, and to show the role of the model theoretic algebraic closure operator for this class of problems. Thomas will study interactions of set theory and group theory in connection with automorphism towers and the cofinalities of infinite analogs of finite groups, making use of forcing, pcf theory, and large cardinals, as well as character theory. An outstanding problem is the relationship of the cofinality of the infinite symmetric group to Blass' invariant, the groupwise density. The role of matrix groups in modern mathematics is well established and occupies a central place in both pure and applied mathematics. It has been conjectured that these groups also play a predominant role in the detailed analysis of many apparently unrelated structures of general type. One of the goals of the present project is to confirm this in the so-called ``tame case'', which is more immediately accessible. A separate goal is to demonstrate that standard ideas of logic (model theory) can cast new light on existing problems in combinatorics (graph theory). It is hoped that graph theorists will themselves adopt these methods in such cases. The work of Thomas uncovers previously unsuspected relations between algebraic and set theoretical issues, and should lead to new ``forcing'' tools, which are among the most powerful foundational tools of modern set theory. The general thrust of the proposal is to show how techniques arising naturally in one area can be transported fruitfully to other subject areas.