Bounded gaps between primes and the length spectra of arithmetic hyperbolic 3-orbifolds
| Autor: | Linowitz, Benjamin, McReynolds, D. B., Pollack, Paul, Thompson, Lola |
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| Rok vydání: | 2017 |
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| Zdroj: | C. R. Math. Acad. Sci. Paris 355 (2017), 1121-1126 |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: V1: 6 pages |
| Databáze: | arXiv |
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