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

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 talk, we demonstrate 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. This talk is based on joint work with B. Linowitz, D. B. McReynolds, and P. Pollack. The talk will be aimed at number theorists; in particular, no geometric background will be assumed.