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Equivariant embeddings of Riemann surfaces in Euclidean spaces with minimal dimensions

Chao Wang, Zhongzi Wang

TL;DR

This work addresses the problem of the smallest dimension $n$ for which a smooth $G$-equivariant embedding of a closed genus $g$ Riemann surface $\Sigma_g$ into $\mathbb{R}^n$ exists, for finite orientation-preserving actions $G\curvearrowright\Sigma_g$. It develops three complementary approaches—group representations, equivariant triangulations, and orbifold theory—to obtain lower bounds and constructive embeddings, proving general upper bounds $d_g(G)\le |G|$ (for $|G|\ge5$) and $d_g(G)\le 12(g-1)$ (for $g\ge2$). A key contribution is the explicit determination of $d_g(G)$ for the first Hurwitz groups arising from principal congruence subgroups: for prime $p\ge7$, $d_g(G)=p+1$ for $(\Sigma(p),\overline{\Gamma}_p)$, and in particular $d_g(G)=8$ for the Klein quartic ($p=7$). The paper also provides explicit embeddings into $\mathbb{R}^{p+1}$ and $\mathbb{C}^{p+1}$, and discusses potential refinements (e.g., relaxing smoothness) and extensions to spherical or minimal-conformal embeddings.

Abstract

Let $Σ_g$ be a closed Riemann surface of genus $g$. Let $G$ be a finite subgroup of the automorphism group of $Σ_g$. It is well known that there exists a smooth $G$-equivariant embedding from $Σ_g$ to some Euclidean space $\mathbb{R}^n$. Let $d_g(G)$ be the minimal possible $n$ for $(Σ_g,G)$. 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 $g\geq 2$.

Equivariant embeddings of Riemann surfaces in Euclidean spaces with minimal dimensions

TL;DR

This work addresses the problem of the smallest dimension for which a smooth -equivariant embedding of a closed genus Riemann surface into exists, for finite orientation-preserving actions . It develops three complementary approaches—group representations, equivariant triangulations, and orbifold theory—to obtain lower bounds and constructive embeddings, proving general upper bounds (for ) and (for ). A key contribution is the explicit determination of for the first Hurwitz groups arising from principal congruence subgroups: for prime , for , and in particular for the Klein quartic (). The paper also provides explicit embeddings into and , and discusses potential refinements (e.g., relaxing smoothness) and extensions to spherical or minimal-conformal embeddings.

Abstract

Let be a closed Riemann surface of genus . Let be a finite subgroup of the automorphism group of . It is well known that there exists a smooth -equivariant embedding from to some Euclidean space . Let be the minimal possible for . We compute the value of 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 , where is prime, . As a corollary, the minimal for the Hurwitz action on the Klein quartic is equal to . 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: if ; and if .

Paper Structure

This paper contains 14 sections, 24 theorems, 66 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Necessary conditions and sufficient conditions
  3. Dimension inequality and eigenvalue condition
  4. Codimension condition and existence theorem
  5. Proofs of Proposition \ref{['thm:Cyclic']} and Theorem \ref{['thm:GEBound']}
  6. Equivariant triangulations and pseudo-triangulations
  7. From equivariant triangulations to equivariant embeddings
  8. Pseudo-triangulations of the $2$-dimensional orbifolds
  9. Large actions and hyperbolic triangle groups
  10. The Riemann surfaces from principal congruence subgroups
  11. Combinatorial construction of the equivariant triangulation
  12. Equivariant embeddings in the complex and real spaces
  13. The minimal equivariant embedding dimension
  14. A concrete real representation of $\mathrm{PSL}(2,\mathbb{Z}/p\mathbb{Z})$

Key Result

Proposition 1.3

For each integer $n\geq 3$, there exist infinitely many $(\Sigma_g,G)$ such that $d_g(G)=n$, where $G$ is cyclic, and it has order $n$ if $n$ is prime.

Figures (9)

  • Figure 1: The generators $\alpha_j$, $\beta_j$, $\gamma_k$ of $\pi_1(\Sigma_g/G,\overline{\star})$.
  • Figure 2: The surface $\Theta$ and the disks $D_0,D_1,\ldots,D_l$.
  • Figure 3: The surface $\Sigma_{1,4}$ and the orbifold $\Sigma_{1,4}/G$.
  • Figure 4: The disks $|\mathrm{St}(v)|$ when $k=3$ and $k=4$.
  • Figure 5: Examples of pseudo-triangles with intersections.
  • ...and 4 more figures

Theorems & Definitions (64)

  • Definition 1.1
  • Proposition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • ...and 54 more