In [Sh771] Shelah rediscovered an old result of Dudley on the non-admissibility of a Polish group topology on an uncountable free group. Crucial to his proof is a so-called Compactness Lemma for Polish groups, concerning satisfaction of algebraic equations for certain sequences of group elements converging to 0 (in distance). In joint work with Shelah we proved that the techniques of [Sh771] apply to many other groups and group constructions, most notably the combinatorial construction known as graph product of groups, a construction that generalizes simultaneously the notion of direct sum and free product of groups. In my talk I will survey our numerous results on the topic and mention the major questions left open by this work.