Model Theory of Nonabelian Free Groups

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This research project centers around model theory and its interactions with hyperbolic geometry. Model theory is concerned with studying structures in a formal mathematical language. One can trace its origins in Gödel’s fundamental work on the completeness of first-order logic. Gödel, in his famous completeness theorem, settled once and for all the question of how truth and provability are connected. Roughly speaking a mathematical statement is provable if and only if it is true in all possible worlds. Although model theory seems to have originated in questions around the philosophy and foundations of mathematics, in recent years profound connections with core mathematical disciplines and computer science have been found. Hyperbolic geometry emerged from the refutation of Euclid’s fifth postulate, also known as the parallel postulate. After more than twenty centuries of efforts to prove the fifth postulate as a consequence of the rest, Lobachevsky and independently Bolyai started developing geometry synthetically employing its negation. This resulted to what is nowadays known as hyperbolic geometry. Although hyperbolic geometry is counterintuitive, in reality space can be curved at places, looking like a horse saddle, and hyperbolic geometry is the right model to apply. This project investigates the connections of model theory with hyperbolic geometry by studying major open questions about the model theory of nonabelian free groups.Nonabelian finitely generated free groups are prototypical examples of hyperbolic groups, i.e. finitely generated groups whose Cayley graph is a hyperbolic metric space. They have attracted much model theoretic attention after the profound result that they share the same first-order theory (Sela, Kharlampovich-Myasnikov). The latter result answered a question of Tarski that remained open for more than fifty years. Maybe surprisingly this common first-order theory is stable. Stability is a tameness condition in Shelah’s classification program and maybe the most prominent dividing line in it. A major aim of this project is the study of natural structures, like fields and groups, interpretable in the first-order theory of nonabelian free groups, which continues prior work of the PI with Ayala Byron and with Chloé Perin, Anand Pillay and Katrin Tent. The first-order theory of nonabelian free groups seems to exhibit some unexpected behavior with respect to the geometry of forking, that will be investigated by the PI. The second component of the project involves the notions of model completeness and model companion. The first-order theory of nonabelian free groups is not model complete, but it might admit a model companion. In the same line of thought properties of existentially closed subgroups of omega residually free groups will be studied. The PI will use techniques from geometric stability theory and geometric group theory. In particular, as demonstrated by previous results of the PI and others, useful tools and notions include, forking calculus, one-basedeness and in general the ample hierarchy, omega residually free towers, test sequences, and the understanding of group actions through Rips machine. A more general goal of the project is to develop methods to tackle similar questions in wider classes of groups, like hyperbolic groups, free products of groups or right-angled Artin groups.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.