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

TL;DR

This paper investigates the minimal asymptotic translation lengths of Torelli groups and pure braid groups on the curve graph, revealing they decay like 1/g and 1/n respectively, contrasting with their larger counterparts.

Contribution

It establishes the asymptotic behavior of translation lengths for Torelli and pure braid groups, showing they decay more slowly than the full mapping class and braid groups.

Findings

Torelli group's translation length behaves like 1/g

Pure braid group's translation length behaves like 1/n

Contrasts with the decay rates of the full groups

Abstract

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 $1/n^2$.