Effective virtual and residual properties of some arithmetic hyperbolic 3-manifolds

TL;DR

This paper establishes effective bounds on subgroup indices and residual finiteness growths for certain arithmetic hyperbolic 3-manifolds, linking geometric properties to algebraic subgroup structures.

Contribution

It provides the first explicit bounds on subgroup indices and residual finiteness growth for specific classes of arithmetic hyperbolic 3-manifolds, extending previous work to non-compact cases.

Findings

Subgroup index is asymptotically smaller than any fractional power of volume.

Effective bounds on geodesic residual finiteness growths are established.

Examples are provided where these bounds apply explicitly.

Abstract

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.