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We study open subgroups $G$ of $\operatorname{SL}_2(\mathbb{Z}_\ell)^n$ interms of some associated Lie algebras without assuming that $G$ is a pro-$\ell$group, thereby extending a theorem of Pink. The result has applications to thestudy of families of Galois representations.

Pink-type results for general subgroup of $\operatorname{SL}_2(\mathbb{Z}_\ell)^n$