Spectrally similar incommensurable 3-manifolds

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.