Equivariant embeddings of Riemann surfaces in Euclidean spaces with minimal dimensions

Let $\Sigma_g$ be a closed Riemann surface of genus $g$. Let $G$ be a finite subgroup of the automorphism group of $\Sigma_g$. It is well known that there exists a smooth $G$-equivariant embedding from $\Sigma_g$ to some Euclidean space $\mathbb{R}^n$. Let $d_g(G)$ be the minimal possible $n$ for $(\Sigma_g,G)$. We compute the value of $d_g(G)$ in certain cases. Especially, we show that: for the automorphism group of the closed Riemann surface which comes from the principal congruence subgroup of level $p$, where $p\geq 7$ is prime, $d_g(G)=p+1$. As a corollary, the minimal $n$ for the Hurwitz action on the Klein quartic is equal to $8$. Three kinds of methods are used in the computation, which are related to the representations of groups, the equivariant triangulations, and the orbifold theory, respectively. The methods are also used to provide two kinds of upper bounds: $d_g(G)\leq |G|$ if $|G|\geq 5$; and $d_g(G)\leq 12(g-1)$ if $g\geq 2$.