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Let $G$ be a compact group. Let $\mathcal N$ be a family of closed normal subgroups of nilpotency class at most $k$. Assume that $\mathcal N$ is closed under finite intersections and $\bigcap_{N\in\...

Let $G$ be a simple (i.e. every proper normal subgroup is discrete) simply connected compact Lie group. Define the degree of $k$-nilpotence of $G$ to be the Haar measure of the set $$\{(x_1,\dotsc,x_{...