Mathematicians are building a unified theory to describe the automorphism groups of Boolean powers—a special class of algebraic structures that includes groups, rings, and Lie algebras. The problem is that most known examples of groups with certain strong combinatorial properties come from relational structures or topological spaces, not from algebraic structures like groups and rings. This project aims to fill that gap by proving that automorphism groups of filtered Boolean powers over the countable atomless Boolean algebra possess two advanced properties: ample generics and extreme amenability. If successful, the research would produce entirely new sources of examples for these properties, drawn from algebra rather than topology or model theory. This is fundamental, curiosity-driven mathematics with no immediate practical application. However, similar work in infinite group theory and Ramsey theory has historically fed into areas as diverse as computer science logic, dynamical systems, and the study of symmetry in complex networks. A deeper understanding of these algebraic automorphism groups could eventually inform how mathematicians and computer scientists think about symmetry, structure, and classification in systems that are too large or infinite to handle directly.
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In a recent piece of work, PL and VR have developed a structure theory for filtered Boolean powers of finite simple non-abelian Malcev algebras and their automorphism groups. Boolean powers are certain kinds of subdirect powers, and play a significant role at the interface of algebra and model theory, as well as in infinite group theory. Malcev algebras are a broad class of algebraic structures, which includes groups, rings, associative algebras, Lie algebras, and loops. Thus they provide a convenient conceptual umbrella within which to prove powerful theorems which then specialise to each of these specific cases. The power of our theory has been demonstrated by showing that when the filtered Boolean power is taken over the countable atomless Boolean algebra, the automorphism group of the resulting structure has some strong desirable combinatorial/model-theoretic properties, namely the Bergman Property and the Small Index Property. In this project we want to deploy the theory further and investigate two even more advanced (and stronger) properties, namely the existence of ample generics and extreme amenability. In doing so, we will need to intertwine our work, with the theories linking these two properties with Fraisse and Ramsey theories (Kechris-Pestov-Todorcevic), as well as the projective Fraisse theory (Irwin-Solecki), and Kwiatkowska's result that the group of autohomemorphisms of the Cantor space has ample generics. A positive outcome in this project would yield new sources of examples of groups possessing these strong properties. These examples would have a hight degree of novelty, because they arise as automorphism groups of algebraic structures (including groups and rings), as opposed to relational structures or autohomeomorphisms of topological spaces, as is overwhelmingly the case in the existing literature. This would include automorphism groups of groups, rings and (classical) algebras. In this way, the proposed research would initiate novel interactions between different mathematical fields and exciting avenues for future research, creating a lasting impact.
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