On commensurability of fibrations on a hyperbolic 3-manifold

TL;DR

This paper explores the relationships between fibrations on hyperbolic 3-manifolds, focusing on commensurability, symmetry, and minimal elements within fibered classes, with new constructions and detailed proofs.

Contribution

It constructs manifolds with non-symmetric but commensurable fibrations and proves conditions under which such fibrations cannot exist, extending understanding of fibered commensurability.

Findings

Constructed manifolds with non-symmetric but commensurable fibrations

Proved that manifolds without hidden symmetries lack non-symmetric commensurable fibrations

Discussed the proof of the minimal element theorem in the cusped case

Abstract

We discuss fibered commensurability of fibrations on a hyperbolic 3-manifold, a notion introduced by Calegari, Sun and Wang. We construct manifolds with non-symmetric but commensurable fibrations on the same fibered face. We also prove that if a given manifold M does not have any hidden symmetries, then M does not admit non-symmetric but commensu- rable fibrations. Finally, Theorem 3.1 of Calegari, Sun and Wang shows that every hyperbolic fibered commensurability class contains a unique minimal element. In this paper we provide a detailed discussion on the proof of the theorem in the cusped case.