Faculty Publications

Publication Date

2017

Disciplines

Geometry and Topology | Mathematics

Abstract

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.

Document Type

Accepted Version

Comments

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).

Rights

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
doi:10.1112/plms.12045

Available for download on Wednesday, August 01, 2018

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