  In this paper, we show that the minimal asymptotic translation length of the
Torelli group $\mathcal{I}_g$ of the surface $S_g$ of genus $g$ on the curve
graph asymptotically behaves like $1/g$, contrary to the mapping class group
$Mod(S_g)$, which behaves like $1/g^2$. We also show that the minimal
asymptotic translation length of the pure braid group $PB_n$ on the curve graph
asymptotically behaves like $1/n$, contrary to the braid group $B_n$, which
behaves like $1/n^2$.
