Abelianizations of finite-index subgroups of the handlebody group

Abstract

For genus $\ge 4$, it is an open question whether the mapping class group of a handlebody contains a finite-index subgroup with nontrivial rational abelianization. In this paper, we provide evidence that no such subgroup exists. First, we prove that, for all such finite-index subgroups $\Gamma$, meridian multitwists vanish in $H_1(\Gamma ;\mathbb{Q})$. Next, we show that $H_1(\Gamma ;\mathbb{Q})=0$ for finite-index subgroups $\Gamma$ containing the handlebody Torelli group, or large enough subgroups of the twist group or the handlebody Johnson kernel.

16 pages, 4 figures