Stallings core graphs are a powerful graphical representation of the finitely generated subgroups of a free group. They bring geometric insight to the study of free groups, make many properties and algorithmic tasks transparent, and are a ubiquitous tool in combinatorial and geometric group theory.

Free group algebras are strikingly analogous to free groups. Most notably, just as the Nielsen-Schreier theorem says that every subgroup of a free group is free, the Cohn-Lewin theorem says that every right (or left) ideal of a free group algebra is free. Yet free group algebras have had no counterpart to core graphs.

In this talk I will present one. We define the core of a module over a free group algebra, and show that it is unique whenever the module is algebraic (also known as bound). Many of the concepts of Stallings' theory, such as folding and pullbacks, generalize to these new objects, and this yields simple proofs and algorithms for many natural problems about free group algebras. For example, I will present an algorithm for testing primitivity in free modules which generalizes (Wilton's simplification of) Whitehead's classical algorithm for testing primitivity in free groups.

This is joint work with Doron Puder.