Covering spaces of 3-orbifolds

Abstract

Let $O$ be a compact orientable $3$-orbifold with nonempty singular locus and a finite volume hyperbolic structure. (Equivalently, the interior of $O$ is the quotient of hyperbolic $3$-space by a lattice in $\mathrm{PSL}(2,\mathbb{C})$ with torsion.) Then we prove that $O$ has a tower of finite-sheeted covers $\{O_i\}$ with linear growth of mod $p$ homology for some prime $p$. This means that the dimension of the first homology, with mod $p$ coefficients, of the fundamental group of $O_i$ grows linearly in the covering degree. The proof combines techniques from $3$-manifold theory with group-theoretic methods, including the Golod-Shafarevich inequality and results about $p$-adic analytic pro-$p$ groups. This has several consequences. First, the fundamental group of $O$ has at least exponential subgroup growth. Second, the covers $\{O_i\}$ have positive Heegaard gradient. Third, we use the existence of this tower of covers to show that a group-theoretic conjecture of Lubotzky and Zelmanov implies that $O$ has a large fundamental group. This implication uses a new theorem of the author, which will appear in a forthcoming paper. These results all provide strong evidence for the conjecture that any closed orientable hyperbolic $3$-orbifold with nonempty singular locus has large fundamental group. Many of these results also apply to $3$-manifolds commensurable with an orientable finite-volume hyperbolic $3$-orbifold with nonempty singular locus. This includes all closed orientable hyperbolic $3$-manifolds with rank-two fundamental group and all arithmetic $3$-manifolds