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

## 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> 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.

## Repository Citation

Linowitz, Benjamin, D.B. McReynolds, Paul Pollack, and Lola Thompson. 2017. “Bounded gaps between primes and the length spectra of arithmetic hyperbolic 3-orbifolds.” Comptes Rendus Mathematique 355(11): 1121-1126.

## Publisher

Elsevier Mason

## Publication Date

11-1-2017

## Publication Title

Comptes Rendus Mathematique

## Department

Mathematics

## Document Type

Article

## DOI

https://dx.doi.org/10.1016/j.crma.2017.07.002

## Language

English

## Format

text