  We give an effective upper bound, for certain arithmetic hyperbolic
3-manifold groups obtained from a quadratic form construction, on the minimal
index of a subgroup that embeds in a fixed 6-dimensional right-angled
reflection group, stabilizing a totally geodesic subspace. In particular, for
manifold groups in any fixed commensurability class we show that the index of
such a subgroup is asymptotically smaller than any fractional power of the
volume of the manifold. We also give effective bounds on the geodesic residual
finiteness growths of closed hyperbolic manifolds that totally geodesically
immerse in non-compact right-angled reflection orbifolds, extending work of the
third author from the compact case. The first result gives examples to which
the second applies, and for these we give explicit bounds on geodesic residual
finiteness growth.
