Finite-index image of a free-group endomorphism is surjective
= Finite-index image of a free-group endomorphism is surjective
An <endomorphism> of $F_n$ has image generated by at most $n$ elements. If this image has finite index $d$, the <Nielsen–Schreier formula> gives rank $1+d(n-1)$. Its <abelianization> requires at least that many generators. For $n\geq2$, comparison with $n$ forces $d=1$.
// Target: geometry-and-topology.bigb