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    Spectrally similar incommensurable 3-manifolds

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    1609.00748v2.pdf
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    Genre
    Pre-print
    Date
    2017-08-01
    Author
    Futer, D
    Millichap, C
    Subject
    57M50
    30F40
    58J53
    53C22
    Permanent link to this record
    http://hdl.handle.net/20.500.12613/4899
    
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    DOI
    10.1112/plms.12045
    Abstract
    © 2017 London Mathematical Society. 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 Nnμ whose volume is approximately n and whose length spectra agree up to length n. Both Nn and Nnμ 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.
    Citation to related work
    Wiley
    Has part
    Proceedings of the London Mathematical Society
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    ae974a485f413a2113503eed53cd6c53
    http://dx.doi.org/10.34944/dspace/4881
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