# Embedding compact surfaces into the 3-dimensional Euclidean space with maximum symmetry

## TL;DR

This paper determines the maximum symmetry group actions on embedded surfaces in three-dimensional space, classifies the topological types that achieve this maximum, and provides explicit embeddings for these highly symmetric surfaces.

## Contribution

It establishes the maximum order of finite symmetry groups acting on embedded surfaces in ^3 and identifies the corresponding topological types and embeddings.

## Key findings

- Maximum order of finite group actions on embedded surfaces in ^3
- Classification of topological types achieving maximum symmetry
- Explicit embeddings for maximally symmetric surfaces

## Abstract

The symmetries of surfaces which can be embedded into the symmetries of the 3-dimensional Euclidean space $\mathbb{R}^3$ are easier to feel by human's intuition. We give the maximum order of finite group actions on $(\mathbb{R}^3, \Sigma)$ among all possible embedded closed/bordered surfaces with given geometric/algebraic genus $>1$ in $\mathbb{R}^3$. We also identify the topological types of the bordered surfaces realizing the maximum order, and find simple representative embeddings for such surfaces.