Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 156 / 1 / b / Solution

Let $G$ be generated by $r$ elements and choose $h\ge \max \{2,r\}$. The fundamental group of a closed orientable surface has the presentationMapping $a_1,\dots ,a_r$ to generators of $G$ and every $b_i$ and remaining $a_i$ to the identity defines a surjective group homomorphism ${\pi }_1(S_h)\twoheadrightarrow G$.

Its kernel determines a connected regular covering $S\to S_h$ with deck transformation group $G$. The action is free and orientation preserving. The covering surface has genus at least two, and the lifted hyperbolic metric makes every deck transformation an isometry. Part a now gives $G\hookrightarrow \mathrm{Mod}(S)$. This is the realization of a finite group as a surface deck group.