On small abstract quotients of Lie groups and locally compact groups

Abstract

Suppose that G is a locally compact group, that \({\Gamma}\) is a discrete, finitely generated group, and that $$\varphi\colon G \longrightarrow \Gamma$$ is an ‘abstract’ surjective homomorphism. We are interested in conditions which imply that \({\varphi}\) is automatically continuous. We obtain a complete answer to this question in the case where G is a topologically finitely generated locally compact abelian group or an almost connected Lie group. In these two cases the well-known structure theory for such groups G leads quickly to a solution. The question becomes much more difficult if one assumes only that G is a locally compact group. This leads to interesting questions about normal subgroups in infinite products and in ultraproducts. Łos’ theorem, the solution of the 5th Hilbert problem, and recent results by Nikolov–Segal can be combined to answer the question.