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

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 $P{B}_n$ on the curve graph asymptotically behaves like $1/n$, contrary to the braid group $B_n$, which behaves like $1/n^2$.

12 pages, 4 figures. The proof of Theorem 5.1 had a minor gap in the previous version which is now fixed. Also the multi-curves in Figure 1 in the previous version were not filling, and this issue has been fixed as well. Accepted for publication in IMRN