Generalized Johnson homomorphisms for extended N-series

TL;DR

This paper develops a generalized theory of Johnson homomorphisms for extended N-series, unifying and extending known variants in the context of group actions on filtered groups.

Contribution

It introduces a broad framework for Johnson homomorphisms applicable to extended N-series, encompassing existing variants and proposing new ones.

Findings

Unified framework for Johnson homomorphisms

Inclusion of known and new variants

Applicable to a wide class of group actions

Abstract

The Johnson filtration of the mapping class group of a compact, oriented surface is the descending series consisting of the kernels of the actions on the nilpotent quotients of the fundamental group of the surface. Each term of the Johnson filtration admits a Johnson homomorphism, whose kernel is the next term in the filtration. In this paper, we consider a general situation where a group acts on a group with a filtration called an "extended N-series". We develop a theory of Johnson homomorphisms in this general setting, including many known variants of the original Johnson homomorphisms as well as several new variants.