Profinite groups are the inverse limits of finite groups, or equivalently, the compact totally disconnected groups. First-order logic in the signature of groups can directly address only their algebraic structure.
We study when a profinite group can be determined by a single first-order sentence within the class of profinite groups. In this case, the algebra determines the topology. We address the analogous question for pro-p groups.
This is joint work with Segal and Tent.
Reference: Finite axiomatizability for profinite groups. Proceedings of the London Mathematical Society 123.6 (2021): 597-635. | 
