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Geometry and Topology | Mathematics


Reid has asked whether hyperbolic manifolds with the same geodesic length spectrum must be commensurable. Building toward a negative answer to this question, we construct examples of hyperbolic 3–manifolds that share an arbitrarily large portion of the length spectrum but are not commensurable. More precisely, for every n ≫ 0, we construct a pair of incommensurable hyperbolic 3–manifolds Nn and Nµn whose volume is approximately n and whose length spectra agree up to length n.

Both Nn and Nµn are built by gluing two standard submanifolds along a complicated pseudo-Anosov map, ensuring that these manifolds have a very thick collar about an essential surface. The two gluing maps differ by a hyper-elliptic involution along this surface. Our proof also involves a new commensurability criterion based on pairs of pants.

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Accepted Version


This article is the author-created version that incorporates referee comments. It is the accepted-for-publication version. The content of this version may be identical to the published version (the version of record) save for value-added elements provided by the publisher (e.g., copy editing, layout changes, or branding consistent with the rest of the publication).


This is the peer reviewed version of the following article: "Spectrally similar incommensurable 3-manifolds," by David Futer and Christian Millichap, which has been published in final form at http://dx.doi.org/10.1112/plms.12045. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.

Original Citation

David Futer & Christian Millichap
Spectrally similar incommensurable 3-manifolds.
Proceedings of the London Mathematical Society, 2017, volume 115, issue 2, pages 411-447

Available for download on Wednesday, August 01, 2018