A small normal generating set for the handlebody subgroup of the Torelli group

TL;DR

This paper proves that the handlebody subgroup of the Torelli group can be generated by genus one BP-maps and provides a normal generating set for related subgroups, advancing understanding of their algebraic structure.

Contribution

It introduces a minimal generating set for the handlebody subgroup of the Torelli group using genus one BP-maps and extends this to the level d mapping class group.

Findings

Handlebody subgroup of Torelli group generated by genus one BP-maps

Normal generating set for handlebody subgroup of level d mapping class group

Advances understanding of subgroup algebraic structures

Abstract

We prove that the handlebody subgroup of the Torelli group of an orientable surface is generated by genus one BP-maps. As an application, we give a normal generating set for the handlebody subgroup of the level $d$ mapping class group of an orientable surface.