# Bounded gaps between primes and the length spectra of arithmetic hyperbolic 3-orbifolds

## TL;DR

This paper demonstrates that in the arithmetic setting, there are infinitely many non-commensurable hyperbolic 3-orbifolds sharing partial geodesic length spectra and bounded volume intervals, using advanced number theory techniques.

## Contribution

It extends bounded gaps results for prime ideals in number fields to the context of hyperbolic 3-orbifolds, showing common spectral features among non-commensurable examples.

## Key findings

- Existence of infinitely many non-commensurable orbifolds with shared length spectra
- Bounded volume intervals for constructed orbifolds
- Application of Chebotarev set prime ideal gaps in geometric context

## Abstract

In 1992, Reid asked whether hyperbolic 3-manifolds with the same geodesic length spectra are necessarily commensurable. While this is known to be true for arithmetic hyperbolic 3-manifolds, the non-arithmetic case is still open. Building towards a negative answer to this question, Futer and Millichap recently constructed infinitely many pairs of non-commensurable, non-arithmetic hyperbolic 3-manifolds which have the same volume and whose length spectra begin with the same first $m$ geodesic lengths. In the present paper, we show that this phenomenon is surprisingly common in the arithmetic setting. In particular, given any arithmetic hyperbolic 3-orbifold derived from a quaternion algebra, any finite subset $S$ of its geodesic length spectrum, and any $k \geq 2$, we produce infinitely many $k$-tuples of arithmetic hyperbolic 3-orbifolds which are pairwise non-commensurable, have geodesic length spectra containing $S$, and have volumes lying in an interval of (universally) bounded length. The main technical ingredient in our proof is a bounded gaps result for prime ideals in number fields lying in Chebotarev sets which extends recent work of Thorner.