Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 156 / 2 / c / Solution

For the punctured torus, $G=⟨\alpha ,\beta ⟩\cong F_2$ and $G^{\mathrm{ab}}\cong {\mathbb{Z}}^2$. Dehn twists about the two standard curves act on homology, in suitable oriented bases, byThe supplied result says that these matrices generate $S{L}_2(\mathbb{Z})$. Thus the image ofcontains the infinite group $S{L}_2(\mathbb{Z})$. Its intermediate group $\mathrm{Out}(F_2)$ must therefore be infinite.