On the Model Theory of the Free Factor Complex

by Zahra Mohammadi Khagheshlaghi (Uni Münster)

Friday Nov 6, 2026, 1:30 PM → 2:30 PM Europe/Berlin

Endenicher Allee 60/1-016 - Lipschitzsaal (Mathezentrum)

The free factor complexes (AFn) and (OFn) encode the structure of free factors of (Fn) and their conjugacy classes. Bestvina and Bridson proved that, for (n≥3), their automorphism groups are precisely Aut(Fn) and Out(Fn), respectively. A key step in their proof is to recognise algebraic relations, such as when free factors give a free product decomposition of (Fn), using only the combinatorics of the complexes. In model theory, this is called definability, and it ensures that automorphisms preserve these relations.

I will explain how this viewpoint leads to a model-theoretic study of these complexes, including how rigidity arguments yield homogeneity and primeness for their first-order theories. The main focus will be stability theory and what it reveals about the complexity of definable relations. In joint work with Katrin Tent, we proved that (AF2), equipped with the complement relation, is (ω)-stable. I will then turn to higher ranks, where (ω)-stability fails, and describe what we know so far about the model theory of these complexes.

Geometry and Groups in North Rhine-Westphalia