Finding fibre faces in finite covers (Q935892)

In this beautiful note the authors prove the following result and corollary: Theorem: Let \(M\) be a closed arithmetic hyperbolic \(3\)-manifold which fibres over the circle. Then given any \(K \in \mathbb{N}\), there exists a finite sheeted covering of \(M\) for which the unit ball of Thurston norm has more than \(K\) fibred faces. Corollary: Let \(M\) be a closed arithmetic hyperbolic \(3\)-manifold which fibres over the circle. Then the rank of the second homology can be increased without bound. The proof of the theorem is purely geometric; the authors apply standard arguments for flows and make use of the arithmeticity of \(M \cong \mathbb{H}^3 / \Gamma\), particularly of the fact that the commensurator of \(\Gamma\) in \(PSL(2,\mathbb{C})\) is dense.