Unshifted rank gradients (${\sup }_{[G:H]<\infty }\frac{d(H)}{[G:H]},{\inf }_{[G:H]<\infty }\frac{d(H)}{[G:H]}$)

These upper and lower quantities use $d(H)$, without subtracting one. For a free group $F_n$ with $n\ge 1$, the Nielsen–Schreier formula gives $d(H)/[F_n:H]=n-1+1/[F_n:H]$. The upper value is $n$, attained at the whole group, and the lower value is $n-1$, approached by kernels of maps onto cyclic groups of arbitrarily large finite order. The frequently used shifted rank gradient replaces $d(H)$ by $d(H)-1$ and therefore gives different upper values.