On commensurability of fibrations on a hyperbolic 3-manifold

We discuss fibered commensurability of fibrations on hyperbolic 3-manifolds, a notion introduced by Calegari, Sun, and Wang (Pacific J. Math. 250:2 (2011), 287-317). We construct manifolds with nonsymmetric but commensurable fibrations on the same fibered face, and prove that if a given manifoldM does not have hidden symmetries, thenM does not admit nonsymmetric but commensurable fibrations. It was also proved by Calegari et al that every hyperbolic fibered commensurability class contains a unique minimal element. Here we provide a detailed discussion on the proof of the theorem in the cusped case.