Abelianizations of finite-index subgroups of the handlebody group

Abstract

For genus $\geq 4$, 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 $Γ$, meridian multitwists vanish in $H_1(Γ; \mathbb{Q})$. Next, we show that $H_1(Γ; \mathbb{Q}) = 0$ for finite-index subgroups $Γ$ containing the handlebody Torelli group, or large enough subgroups of the twist group or the handlebody Johnson kernel.