Every group is the automorphism group of a graph with arbitrarily large genus

We prove that, to every abstract group $G$, we can associate a sequence of graphs ${\mathrm{\Gamma }}_n$ such that the automorphism group of ${\mathrm{\Gamma }}_n$ is isomorphic to $G$ and the genus of ${\mathrm{\Gamma }}_n$ is an unbounded function of $n$.