Density Spectra of Topological Groups

This paper investigates the density spectra of topological groups, focusing on the contrasting topological behaviors of dense subgroups and closed subgroups. For dense subgroups, we study the density spectrum $\dd^*(G)$ and the conjecture that every compact group satisfies property $\JM^*$, namely $\dd^*(G) = [d(G), w(G)]$. We establish a structural reduction, proving that the conjecture holds for all compact groups if it can be verified that the upper bound $w(Q) \in \dd^*(Q)$ is satisfied by all profinite groups $Q$. Utilizing this reduction, we confirm the conjecture for pronilpotent groups. For closed subgroups, we analyze the closed density spectrum $\cd(G)$ and resolve two notable open problems. First, we provide an affirmative answer in $\mathbf{ZFC}$ to a question posed by Leiderman, Morris, and Tkachenko by constructing a separable countably compact Boolean group that contains a closed non-separable subgroup. Second, we resolve a problem of Hern\'andez, Hofmann, and Morris in the negative, shwoing that there exist profinite groups without any non-trivial metrizable closed normal subgroups.