Minimal asymptotic translation lengths of Torelli groups and pure braid groups on the curve graph

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 \(1/n^2\).