Double Johnson filtrations for mapping class groups

TL;DR

This paper develops a comprehensive framework for Johnson filtrations and homomorphisms in group actions, introducing double Johnson filtrations and applying them to mapping class groups and automorphism groups of free groups.

Contribution

It generalizes Johnson filtrations to a bi-graded setting and applies this theory to mapping class groups and automorphism groups, revealing new structural insights.

Findings

Established a general theory of Johnson filtrations for groups acting on filtered groups.

Introduced double Johnson filtrations and homomorphisms for bi-graded filtrations.

Applied the theory to mapping class groups and automorphism groups, deriving new algebraic structures.

Abstract

We first develop a general theory of Johnson filtrations and Johnson homomorphisms for a group $G$ acting on another group $K$ equipped with a filtration indexed by a "good" ordered commutative monoid. Then, specializing it to the case where the monoid is the additive monoid $\mathbb{N}^2$ of pairs on nonnegative integers, we obtain a theory of double Johnson filtrations and homomorphisms. We apply this theory to the mapping class group $\mathcal{M}$ of a surface $\Sigma_{g,1}$ with one boundary component, equipped with the normal subgroups $\bar{X}$, $\bar{Y}$ of $\pi_1(\Sigma_{g,1})$ associated to a standard Heegaard splitting of the $3$-sphere. We also consider the case where the group $G$ is the automorphism group of a free group.