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Let $G$ be a locally compact group and let $H$ be a cocompact (or more generally, a cofinite) subgroup of $G$.

Is there any criterion to determine whether $H$ contains a cocompact normal subgroup of $G$? More specifically, when does $G$ possess a normal cocompact subgroup? (regardless of $H$)

Is there a non-obvious example (e.g. $G$ non-abelian and non-compact) of this phenomenon?

P.S. Feel free to restrict your answer to Lie groups if it helps.

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A necessary condition is that there exists on $G/H$ an invariant Radon measure, or equivalently that the modular function of $G$ extends that of $H$. (For instance this fails if $G$ is unimodular but not $H$, e.g. upper triangular matrices in $SL_{n\ge 2}(\mathbf{R})$) –  YCor Sep 2 '13 at 9:33

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