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

Abstract

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

A classification of maximally symmetric surfaces in the 3-dimensional torus

“…The most natural choices of M include the 3-dimensional Euclidean space R 3 and the 3-dimensional sphere S 3 . In each case, the classification does exist, and it is stronger in the sense that for each given g the maximum of the group order can be obtained and the actions realizing the maximum can be classified (see [WWZZ3] for R 3 and [WWZZ1,WWZZ2] for S 3 ).…”

“…Conversely, given a 2-orbifold F in a 3-orbifold O and a regular orbifold covering p : M → O, if p −1 (F ) is connected, then the group π 1 (O)/π 1 (M ) acts on the pair (M, p −1 (F )). If M is R 3 or S 3 , then finding all the pairs (O, F ) is enough (as in [WWZZ2,WWZZ3]), because R 3 and S 3 are simply connected and p is determined by O. For T 3 further information about the covering p is needed.…”

The minimal dimension of a sphere with an equivariant embedding of the bouquet of $g$ circles is $2g-1$

“…A related question is when a finite group action G on a graph Γ can be G-equivariantly embedded into R 3 (S 3 ), see [FNPT], [WWZZ2]. Similar question for surfaces is also addressed, see [CC], [WWZZ1].…”