On homomorphic images of direct products

Previously at this seminar, I was talking about how an embedding into a direct product of algebraic systems could be factored through an ultraproduct. This time I will tell about a dual version of this, with embedding replaced by homomorphism. The motivation comes from a recent model-theoretic approach to Quantum Mechanics due to Zilber. A central open question in this approach is whether one can map an ultraproduct of finite groups into some physically relevant Lie groups, such as SO(3). I will show how this question easily reduces to the same question about direct products. The latter question seems to be difficult. I will discuss it under various viewpoints, including a universal-algebraic one, and, if time will permit, will observe what is known in that regard (not much): finitenss of a countably-generated image of a profinite group (Nikolov and Segal), structure of simple pseudo-finite groups (J.S. Wilson).