Residual Finiteness Growths of Closed Hyperbolic Manifolds in Non-compact Right-angled Reflection Orbifolds

We bound the geodesic residual finiteness growths of closed hyperbolic manifolds that immerse totally geodesically in non-compact right-angled reflection orbifolds, extending work of the fourth author from the compact case. We describe a rich class of arithmetic examples to which this result applies, and apply the theorem to give effective bounds on the geodesic residual finiteness growths of members of this family. This paper contributes a few observations to the project of quantifying residual finiteness growth. Our first main result, Theorem 1.6, extends those of [16], which established explicit linear bounds on the geodesic residual finiteness growth of fundamental groups of closed hyperbolic manifolds that admit totally geodesic immersions to a compact right-angled reflection orbifolds. Our results still concern closed manifolds, but we allow non-compact right-angled reflection orbifolds of finite volume. Our bound depends on a choice of “embedded” horoballs. Definition 1. For a polyhedron P and an ideal vertex v of P , we will say a horoball centered at v is embedded in P if it does not intersect the interior of any side of P that is not incident on v. Here and below, the term “side” of a polyhedron P refers specifically to a codimension-one face of P , following Ratcliffe (see [19], p. 198 and Theorem 6.3.1). Theorem 1.6. For n ≥ 2, let P be a right-angled polyhedron in H with finite volume and at least one ideal vertex, let ΓP be the group generated by reflections in the sides of P , and let B be a collection of horoballs, one for each ideal vertex of P , that are each embedded in the sense of Definition 1 and pairwise non-overlapping. For a closed hyperbolic m-manifold M , m ≤ n, that admits a totally geodesic immersion to OP . = H/ΓP , and any α ∈ π1M −{1}, there exists a subgroup H ′ of π1M such that α / ∈ H ′, and the index of H ′ is bounded above by 2vn(1) VR+hmax sinh (R+ dR+hmax) `(α), where vn(1) is the (Euclidean) volume of the n-dimensional Euclidean unit ball and: • `(α) is the length of the unique geodesic representative of α; • R = ln( √ n+ 1 + √ n); • hmax = ln(cosh rmax), where rmax is the radius of the largest embedded ball in M ; and • dR+hmax and VR+hmax are the diameter and volume, respectively, of the (R+ hmax)-neighborhood in P of P − ⋃ {B ∈ B}.