Finiteness properties of the Johnson subgroups

TL;DR

This paper investigates the finiteness of the first rational homology of Johnson subgroups of the mapping class group, providing evidence of finite-dimensionality after completion for genus g > 3, and exploring homology properties of the Johnson filtration.

Contribution

It demonstrates the finite-dimensionality of the completed first rational homology of Johnson subgroups for large genus and shows infinite-dimensionality in some degrees of the Johnson filtration's homology.

Findings

Finite-dimensionality of the completed H_1(K_{g,1},Q) for g > 3.

Infinite-dimensional rational homology in some degrees of Johnson filtration.

Extension of previous results to almost all genera.

Abstract

The main goal of this note is to provide evidence that the first rational homology of the Johnson subgroup $K_{g,1}$ of the mapping class group of a genus g surface with one marked point is finite-dimensional. Building on work of Dimca-Papadima, we use symplectic representation theory to show that, for all $g > 3$, the completion of $H_1(K_{g,1},\mathbb{Q})$ with respect to the augmentation ideal in the rational group algebra of $\mathbb{Z}^{2g}$ is finite-dimensional. We also show that the terms of the Johnson filtration of the mapping class group have infinite-dimensional rational homology in some degrees in almost all genera, generalizing a result of Akita.