Bounded gaps between primes and volumes of manifolds

In 1992, Reid posed the question of whether hyperbolic 2- and 3-manifolds with the same geodesic length spectra are necessarily commensurable. Reid subsequently answered this question in the arithmetic setting; the non-arithmetic case remains open. In this talk, we give an effective version of Reid's results, showing that, if the geodesic lengths agree up to a certain bound, then a pair of arithmetic hyperbolic 2- or 3- manifolds are necessarily commensurable. At the same time, we show that there are lots of pairwise non-commensurable arithmetic hyperbolic 2- and 3-orbifolds with a great deal of overlap in their geodesic lengths. In fact, it turns out that there are infinitely many k-tuples of such orbifolds with volumes lying in an interval of bounded length. Because of the correspondence between maximal subfields of quaternion algebras and geodesics on arithmetic hyperbolic manifolds, the main tools used in these proofs come from analytic number theory. In particular, one of the key ideas stems from the breakthrough work of Maynard and Tao on bounded gaps between primes. This talk is based on a series of joint papers with Benjamin Linowitz, D. B. McReynolds, and Paul Pollack.