= Unshifted rank gradients
{title2=$\sup_{[G:H]<\infty}\frac{d(H)}{[G:H]},\quad\inf_{[G:H]<\infty}\frac{d(H)}{[G:H]}$}
These upper and lower quantities use $d(H)$, without subtracting one. For a <free group> $F_n$ with $n\ge1$, 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.
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