Maximum orders of extendable actions on surfaces

Abstract

We determine the maximum order E g of finite groups G acting on the closed surface Σ g of genus g which extends over (S 3 , Σ g ) for all possible embeddings Σ g → S 3 , where g > 1.

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

“…On the other hand, the embedded surfaces realizing E g must be unknotted. At this point, it would be worth to compare with those results in [WWZZ2] and [WWZ1].…”

“…(2) Maximum orders of finite group actions on surfaces which can extend over a given 3-manifold M : (i) The maximum order of finite group actions on Σ g with g > 1 is at most 12(g−1) when M is a handlebody and Σ g = ∂M , [Zi] in 1979. (ii) Much more recently the maximum order of extendable finite group actions on Σ g is determined when M is the 3-sphere S 3 , for cyclic group, see [WWZZ1] for orientation-preserving case and [WZh] for general case; for general finite groups, see [WWZZ2] for orientation-preserving case and [WWZ1] for general case. (iii) Some progress has been made when M is the 3-torus T 3 , see [WWZ2] for cyclic case and [BRWW] for general case.…”

The maximally symmetric surfaces in the 3-torus

“…A similar problem has been addressed for surfaces embedded in the 3sphere, S 3 , which is the simplest compact 3-manifold in the sense that it is a one point compactification of our three space and the universal spherical 3-manifold covering all spherical 3-manifolds. See [WWZZ1] and [WWZZ2] for surfaces in the orientable category. Note that only orientable surfaces Σ g 2010 Mathematics Subject Classification.…”

Graphs in the 3-Sphere with Maximum Symmetry

“…Notice that generally ∂N ǫ (Γ) is not smooth. But we can choose a smaller equivariant neighborhood U ǫ of Γ such that ∂U ǫ is a smooth submanifold in S 3 and ∂U ǫ ≃ Σ g (see [WWZ,Remark 3.3]). Hence we have a G-action on (S 3 , Σ g ), and it follows that m g ≤ OE g .…”