Residual finiteness for locally indicable groups 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 $G$?

Any non-residually finite, torsion-free one-relator group is also a counter-example. Such a group maps on to $\mathbb{Z}$ so has normal subgroups of each finite index, and is locally indictable by a result of Jim Howie*.
For example, take $G=\langle a, b; a^{-1}b^2a=b^3\rangle$.
*Howie, J. On locally indicable groups, Math. Z. (1982) 180: 445. https://doi.org/10.1007/BF01214717
