Double Johnson filtrations for mapping class groups

Abstract

In this paper, we first develop a general theory of Johnson filtrations and Johnson homomorphisms for a group [Formula: see text] acting on another group [Formula: see text] equipped with a filtration indexed by a “good” ordered commutative monoid. Then, specializing it to the case where the monoid is the additive monoid [Formula: see text] of pairs on non-negative integers, we obtain a theory of double Johnson filtrations and homomorphisms. We apply this theory to the mapping class group [Formula: see text] of a surface [Formula: see text] with one boundary component, equipped with the normal subgroups [Formula: see text], [Formula: see text] of [Formula: see text] associated to a standard Heegaard splitting of the [Formula: see text]-sphere. We also consider the case where the group [Formula: see text] is the automorphism group of a free group.