Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 141 / 2 / e / Solution

The matrix in part (d) has trace $4$ and determinant $1$. Since $|\mathrm{tr}{\varphi }_{\ast }|>2$, the once-punctured-torus case of the Nielsen–Thurston classification theorem says that $\varphi$ is pseudo-Anosov. The Hyperbolization of a pseudo-Anosov mapping torus therefore gives