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Let $G$ be a finitely generated locally indicable group. For every $n \in \mathbb{N} $ there is a normal subgroup with index $n$. What can we say about residual finiteness of $G$?

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Thompson's group $F$ has a normal subgroup of each finite index, just because it admits $\mathbf{Z}$ as a quotient, it is locally indicable (= every nontrivial f.g. subgroup admits $\mathbf{Z}$ as a quotient) but $F$ is not residually finite.

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