Take the three-punctured sphere $S_{0,3}$. Its full mapping class group is finite, isomorphic to the permutation group $S_3$, whereas
$$
\pi_1(S_{0,3})\cong F_2
$$
and $\operatorname{Out}(F_2)$ is infinite by part c. Hence the finite subgroup $p(\operatorname{Mod}(S_{0,3}))$ has infinite index in $\operatorname{Out}(\pi_1(S_{0,3}))$.


Solution

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

 Solution

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