Bounded gaps between primes and the length spectra of arithmetic hyperbolic 3-orbifolds |
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Authors: | Benjamin Linowitz D.B. McReynolds Paul Pollack Lola Thompson |
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Affiliation: | 1. Department of Mathematics, Oberlin College, Oberlin, OH 44074, USA;2. Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA;3. Department of Mathematics, University of Georgia, Athens, GA 30602, USA |
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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 , 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. |
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