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Among compact Hausdorff groups $G$ whose maximal profinite quotient is finitely generated, we characterize those that possess a proper dense normal subgroup. We also prove that the abstract commutator subgroup $[H,G]$ is closed for every closed normal subgroup $H$ of $G$.

For a group $G$ and a positive real number $x$, define $d_G\left(x\right)$ to be the number of integers less than $x$ which are dimensions of irreducible complex representations of $G$. We study the asymptotics of $d_G\left(x\right)$ for algebraic groups, arithmetic groups and finitely generated linear groups. In particular we prove an “alternative” for finitely generated linear groups $G$ in characteristic zero, showing that either there exists $\alpha >0$ such that $d_G\left(x\right)>x^{\alpha }$ for all large $x$, or $G$ is virtually abelian (in which case $d_G\left(x\right)$ is bounded).