An asymptotic local-global theorem on heights of some Kleinian group orbits

TL;DR

This paper proves an asymptotic local-global theorem concerning the heights of point orbits of thin subgroups of Bianchi groups in hyperbolic 3-space using the circle method.

Contribution

It introduces a novel application of the circle method to establish asymptotic local-global results for heights in Kleinian group orbits.

Findings

Established an asymptotic formula for heights of orbits

Applied circle method to Kleinian groups

Extended local-global principles to hyperbolic geometry

Abstract

We use circle method prove an asymptotic local-global theorem on the heights of point orbits of thin subgroups of Bianchi groups in $\mathbb H^3$.