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