Spectrally similar incommensurable 3-manifolds

TL;DR

This paper constructs pairs of hyperbolic 3-manifolds that share large portions of their length spectra but are not commensurable, challenging a question by Reid about spectral determination of manifold commensurability.

Contribution

It provides explicit examples of incommensurable hyperbolic 3-manifolds with arbitrarily large common length spectra, using novel gluing techniques and a new commensurability criterion.

Findings

Constructed incommensurable manifolds sharing length spectra up to length n

Developed a new commensurability criterion based on pairs of pants

Manifolds have large volume and thick collars around essential surfaces

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 all sufficiently large n, we construct a pair of incommensurable hyperbolic 3-manifolds $N_n$ and $N_n^\mu$ whose volume is approximately n and whose length spectra agree up to length n. Both $N_n$ and $N_n^\mu$ 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.