A lower bound of the distortion of the Torelli group in the mapping class group with boundary components

TL;DR

This paper establishes that the Torelli group is exponentially distorted within the mapping class group for surfaces with boundary, and shows that this distortion remains consistent across different level structures.

Contribution

It introduces a proof that the Torelli group's distortion is exponential and demonstrates its invariance across level d mapping class groups.

Findings

Torelli group is exponentially distorted in the mapping class group.

Distortion remains the same in level d mapping class groups.

Uses Broaddus-Farb-Putman's techniques for proof.

Abstract

We prove that each Torelli group of an orientable surface with any number of boundary components is at least exponentially distorted in the mapping class group by using Broaddus-Farb-Putman's techniques. Further we show that the distortion of each Torelli group in the level $d$ mapping class group is the same as that of in the mapping class group.