Spectrally similar incommensurable 3‐manifolds. Issue 2 (11th May 2017)

• Record Type: Journal Article
• Authors: Futer, David; Millichap, Christian
• Is Part Of: Proceedings of the London Mathematical Society. Volume 115:Issue 2(2017)
• Page Start: 411
• Page End: 447
• Publication Date: 2017-05-11
• Subjects: 57M50 -- 30F40 -- 58J53 -- 53C22; Mathematics -- Periodicals; Mathematics; Periodicals; 510
• DOI: 10.1112/plms.12045
• Languages: English
• ISSNs: 0024-6115

Abstract

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 N n and N n μ whose volume is approximately n and whose length spectra agree up to length n . Both N n and N n μ 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.