The Involution Rank of $\text{MCG}({\mathrm{\Sigma }}_3)$ Is Four

For the closed orientable surface ${\mathrm{\Sigma }}_3$, we prove $$\text{irank}\text{MCG}({\mathrm{\Sigma }}_3)=4.$$ Thus $\text{MCG}({\mathrm{\Sigma }}_3)$ can be generated by four involutions, but not by three. This settles the genus-three case of Farb’s problem on generation by involutions.