Let $G$ be generated by $r$ elements and choose $h\geq\max\{2,r\}$. The <fundamental group of a closed orientable surface> has the presentation
$$
\pi_1(S_h)=
\langle a_1,b_1,\ldots,a_h,b_h\mid
\textstyle\prod_{i=1}^h[a_i,b_i]=1\rangle.
$$
Mapping $a_1,\ldots,a_r$ 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 $G\hookrightarrow\operatorname{Mod}(S)$. This is the <realization of a finite group as a surface deck group>.

Solution

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

Every finite group $G$ is the deck group of a connected finite regular covering of some closed orientable surface of genus at least two. Choose a surface group that surjects onto $G$ and take the covering corresponding to the kernel. The resulting free action embeds $G$ into the mapping class group of the covering surface.


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