On the top homology group of Johnson kernel

TL;DR

This paper investigates the top homology group of the Johnson kernel, showing it is infinitely generated and contains a free abelian subgroup of infinite rank, revealing complex algebraic structures in the mapping class group's subgroups.

Contribution

It proves that the top homology group of the Johnson kernel is infinitely generated and contains an infinite rank free abelian subgroup, a novel insight into its algebraic structure.

Findings

The top homology group $H_{2g-3}(\\mathcal{K}_g)$ is not finitely generated.

$H_{2g-3}(\\mathcal{K}_g,\ ext{Q})$ contains an infinite rank free abelian subgroup.

$H_{2g-3}(\\mathcal{K}_g,\ ext{Q})$ is not finitely generated as a module over $ ext{Q}[\\mathcal{I}_g]$.

Abstract

The action of the mapping class group $\mathrm{Mod}_g$ of an oriented surface $\Sigma_g$ on the lower central series of $\pi_1(\Sigma_g)$ defines the descending filtration in $\mathrm{Mod}_g$ called the Johnson filtration. The first two terms of it are the Torelli group $\mathcal{I}_g$ and the Johnson kernel $\mathcal{K}_g$. By a fundamental result of Johnson (1985), $\mathcal{K}_g$ is the subgroup of $\mathrm{Mod}_g$ generated by all Dehn twists about separating curves. In 2007, Bestvina, Bux, and Margalit showed the group $\mathcal{K}_g$ has cohomological dimension $2g-3$. We prove that the top homology group $H_{2g-3}(\mathcal{K}_g)$ is not finitely generated. In fact, we show that it contains a free abelian subgroup of infinite rank, hence, the vector space $H_{2g-3}(\mathcal{K}_g,\mathbb{Q})$ is infinite-dimensional. Moreover, we prove that $H_{2g-3}(\mathcal{K}_g,\mathbb{Q})$ is not finitely generated as a module over the group ring $\mathbb{Q}[\mathcal{I}_g]$.