Title: Subgroup separability, retracts and (weakly) branch groups.

Abstract:
 

The notion of separability arose from the study of covering spaces. If G is the fundamental group of a space X, a finitely generated subgroup H determines a covering space X_H of X. Subgroup separability asks when a compact subspace L of X_H embeds into some finite cover K of X; in this case, H is said to be separable in G.

In the classical examples-such as free groups, surface groups, and limit groups-the embedding of L into K often induces a retract at the level of fundamental groups, motivating the study of (virtual) retracts in groups.

The discovery of (weakly) branch groups introduced groups with first-found phenomena not present in classical settings, requiring new techniques for their study. In my talk, I will show how the expected connection between subgroup separability and retracts behaves very differently in branch groups and their generalizations.