For the punctured torus, $G=\langle\alpha,\beta\rangle\cong F_2$ and $G^{\mathrm{ab}}\cong\mathbb Z^2$. Dehn twists about the two standard curves act on homology, in suitable oriented bases, by
$$
\begin{pmatrix}1&0\\1&1\end{pmatrix},
\qquad
\begin{pmatrix}1&-1\\0&1\end{pmatrix}.
$$
The supplied result says that these matrices generate $SL_2(\mathbb Z)$. Thus the image of
$$
q\circ p:\operatorname{Mod}(T_*^2)\longrightarrow GL_2(\mathbb Z)
$$
contains the infinite group $SL_2(\mathbb Z)$. Its intermediate group $\operatorname{Out}(F_2)$ must therefore be infinite.


Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-156/2/c/solution