Double Johnson filtrations for mapping class groups

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 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 M of a surface Σg,1 with one boundary component, equipped with the normal subgroups X, Ȳ of π1(Σg,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.

Generalized Johnson homomorphisms for extended N-series

“…We consider here Oda's embeddings [Oda92] of the g-strand pure braid group P B g into the mapping class group M = M(Σ, ∂Σ) and, to fit our purposes, we assume here that the image of the embedding is contained in the twist group T = T (V ). Embeddings of the (framed) pure braid groups into the twist groups, in the context of Johnson filtrations, were also considered in [HV20].…”