A new filtration of the Magnus kernel of the Torelli group

TL;DR

This paper introduces a new filtration of the Magnus kernel of the Torelli group, revealing its complex structure and establishing connections to lower central series quotients and abelian groups through Johnson-type homomorphisms.

Contribution

It defines an infinite filtration of the Magnus kernel and develops methods to generate nontrivial elements, uncovering its rich algebraic structure and surjective properties.

Findings

The kernel of the Magnus representation is highly non-trivial and structured.

Each filtration quotient contains a subgroup isomorphic to a lower central series quotient.

The quotients surject onto infinite rank torsion-free abelian groups for genus greater than 2.

Abstract

For a oriented genus g surface with one boundary component, S, the Torelli group is the group of orientation preserving homeomorphisms of S that induce the identity on homology. The Magnus representation of the Torelli group represents the action on F/F" where F=pi_1(S) and F" is the second term of the derived series. We show that the kernel of the Magnus representation, Mag(S), is highly non-trivial and has a rich structure as a group. Specifically, we define an infinite filtration of Mag(S) by subgroups, called the higher order Magnus subgroups, M_k(S). We develop methods for generating nontrivial mapping classes in M_k(S) for all k and g>1. We show that for each k the quotient M_k(S)/M_{k+1}(S) contains a subgroup isomorphic to a lower central series quotient of free groups E(g-1)_k/E(g-1)_{k+1}. Finally We show that for g>2 the quotient M_k(S)/M_{k+1}(S) surjects onto an infinite rank torsion free abelian group. To do this, we define a Johnson-type homomorphism on each higher order Magnus subgroup quotient and show it has a highly non-trivial image.