Residually finite group with one subgroup of each finite index is infinite cyclic

The epimorphism $S\to S_{\mathrm{ab}}\cong \mathbb{Z}$ has preimage of $n\mathbb{Z}$ as the unique index-$n$ subgroup. Thus every finite-index subgroup contains the commutator subgroup. Residual finiteness makes their intersection trivial, forcing the commutator subgroup to vanish and $S\cong \mathbb{Z}$.

// Target: geometry-and-topology.bigb