On anomalous subvarieties of holonomy varieties of hyperbolic 3-manifolds

Abstract

The goal of this paper is to explore the interplay between two seemingly distinct fields. More precisely, let $\mathcal{M}$ be an $n$-cusped hyperbolic $3$-manifold with rationally independent cusp shapes, and $\mathcal{X}$ be its holonomy variety. We study the structure of anomalous subvarieties of $\mathcal{X}$, a concept originating in arithmetic geometry, and relate it to various geometric properties of $\mathcal{M}$. First, we show that every maximal anomalous subvariety of $\mathcal{X}$ containing the identity is its subvariety of codimension $1$ which arises by keeping one cusp of $\mathcal{M}$ complete. Second, we show that, if $\mathcal{X}$ is degenerated by its anomalous subvarieties (i.e., $\mathcal{X}^{oa}=\emptyset$), then $\mathcal{M}$ has cusps which are, while keeping some other cusps of it complete, strongly geometrically isolated from the rest. Finally, we completely classify and characterize the case $\mathcal{X}^{oa}=\emptyset$ for the holonomy variety $\mathcal{X}$ of any $2$-cusped hyperbolic $3$-manifold.