Abstracts Mathematics

Model theory and rough approximate subgroups

by Arturo Rodriguez Fanlo

The aim of this thesis is to use model theory to study rough approximate subgroups, generalising various known results about approximate subgroups. Firstly, a systematic study of the structure of piecewise hyperdefinable sets is developed. In particular, we show the most significant properties of their logic topologies. Then, we study piecewise hyperdefinable groups, generalising and improving two of the main model-theoretic results of "Stable group theory and approximate subgroups" by E. Hrushovski. The first one is the existence of Lie models. The second one is the Stabilizer Theorem. In the process, we define the model theoretic components for piecewise hyperdefinable groups, and introduce a new component. We use these results to generalise the Lie Model Theorem, one of the main applications of model theory to study approximate subgroups, to the case of rough approximate subgroups. Secondly, we focus on the case of metric approximate subgroups, i.e. rough approximate subgroups in metric groups. In this particular case, we show that ultraproducts of metric approximate subgroups satisfying some discretisation conditions have Lie models up to an infinitesimal thickening. We conclude using this result to get various combinatorial consequences.