arXiv subjects

Annie Holden

Publications and source records attributed to Annie Holden.

4 recordsLinked to original sources

arXiv2026

The Birman--Craggs--Johnson homomorphism and the handlebody Torelli group

We use the Birman--Craggs--Johnson (BCJ) homomorphism to study the intersection of the handlebody group with the Torelli group and with the Johnson kernel. For genus $\geq 3$, we determine the images of the BCJ homomorphism restricted to these subgroups. We then explicitly compute cup products of pairs of cohomology classes in $H^1(-;\mathbb{F}_2)$ detected by the BCJ homomorphism for genus $\geq 4$. For the handlebody Johnson kernel, our results lift to integral coefficients.

math.GT↗

arXiv2026

Abelianizations of finite-index subgroups of the handlebody group

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.

math.GT↗

arXiv2025

Johnson homomorphisms and the second rational cohomology of handlebody Torelli groups

We introduce two Torelli subgroups of the handlebody group. The group $HI_{g,p}^b$ is the subgroup of the handlebody group acting trivially on the first homology of the boundary surface, and $H_B I_{g,p}^b$ is the subgroup of the handlebody group acting trivially on the first homology of the handlebody. Using the symplectic representation and the Johnson homomorphisms for the Torelli subgroups of the mapping class group and of $\operatorname{Aut}(F_g)$, we define abelian quotients of these handlebody Torelli groups. In terms of the representation theory of the special linear group, we describe cup products of two classes in the first rational cohomology groups of $HI_{g,p}^b$ and $H_B I_{g,p}^b$ obtained by the rational duals of these abelian quotients.

math.GT↗

arXiv2020

On quotients of congruence subgroups of braid groups

The integral Burau representation provides a map from the braid group into a group of integral matrices. This allows for a definition of congruence subgroups of the braid group as the preimage of the usual principal congruence subgroups of integral matrices. We explore the structure these congruence subgroups by examining some of the quotients that may arise in the series induced by divisibility of levels. We build on the work of Stylianakis on symmetric quotients of congruence subgroups, which itself generalizes the quotient of the braid group by the pure braid group. We accomplish this by utilizing results of Newman on integral matrices and explicitly finding elements in the preimage of any transposition. Our generalization is made possible by avoiding the use of a generating set for congruence subgroups. We find further generalizations based on results of Brendle and Margalit as well as Kordek and Margalit on the level four congruence subgroup. This gives families of quotients which are not isomorphic to symmetric groups.

math.GR↗