Title: Caley graphs and Finite Axiomatizability
Speaker: Léo Jimenez (The Fields Institute and The University of Waterloo)
Speaker's URL: https://www.ljimenezmath.com/
Abstract: In model theory, finding stable, superstable, or omega stable finitely axiomatizable theories required some combinatorial creativity. Curiously, no strongly minimal, finitely axiomatizable theory is known. It was shown by Hrushovski that an example must be locally modular, which allows the problem to be divided into the trivial and non-trivial case. In the 90s, Ivanov made a lot of progress regarding the trivial case by examining the specific case of Caley graphs. The Caley graph of a finitely generated group is always strongly minimal, and Ivanov identified a necessary and sufficient condition for its finite axiomatizability. Unfortunately, the existence of groups satisfying this condition is unknown, and connected to difficult group theory problems, such as the existence of a finitely presented group with finitely many conjugacy classes. In this talk, I will give an alternative exposition and proof of Ivanov's result, and discuss adjacent problems. This is joint work with David Meretzky, Rachel Skipper and Caroline Terry.