Cutting and Pasting in the Torelli subgroup of Out($F_n$)

TL;DR

This paper investigates the Torelli subgroups of automorphism groups of free groups with boundary, establishing their finite generation and their adherence to a version of the Birman exact sequence, using 3-manifold inspired techniques.

Contribution

It introduces the first finite generation results and a Birman exact sequence analogue for Torelli subgroups of automorphism groups of free groups with boundary.

Findings

Torelli subgroups are finitely generated.

They satisfy a version of the Birman exact sequence.

Methods adapt ideas from 3-manifold topology and surface mapping class groups.

Abstract

Using ideas from 3-manifolds, Hatcher--Wahl defined a notion of automorphism groups of free groups with boundary. We study their Torelli subgroups, adapting ideas introduced by Putman for surface mapping class groups. Our main results show that these groups are finitely generated, and also that they satisfy an appropriate version of the Birman exact sequence.