On anomalous subvarieties of holonomy varieties of hyperbolic 3-manifolds

Abstract

Let $M$ be an $n$-cusped hyperbolic $3$-manifold having rationally independent cusp shapes and $X$ be its holonomy variety. We first show that every maximal anomalous subvariety of $X$ containing the identity is its subvariety of codimension $1$ which arises by having a cusp of $M$ complete. Second, we prove if $X^{oa} =\emptyset$ , then $M$ has cusps which are, keeping some other cusps of it complete, strongly geometrically isolated from the rest. Third, we resolve the Zilber-Pink conjecture for holonomy varieties of any $2$-cusped hyperbolic $3$-manifolds.