Rank-to-index limit for subgroups of a free group ($\mathrm{rank}(H)/[F_n:H]=n-1+1/[F_n:H]$)

For finite-index subgroups of a rank-$n$ free group, the Nielsen–Schreier formula gives the displayed quotient. Along any sequence whose finite indices tend to infinity, the quotient tends to $n-1$. Subtracting one from the rank before dividing gives the constant rank gradient $n-1$.

// Target: geometry-and-topology.bigb