2
$\begingroup$

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 $G$?

2 Answers 2

2

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.

0

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