Maximum Orders of Cyclic and Abelian Extendable Actions on Surfaces

Abstract

Let ${\Sigma }_g(g>1)$ be a closed surface embedded in $S^3$. If a group $G$ can acts on the pair $(S^3,{\Sigma }_g)$, then we call such a group action on ${\Sigma }_g$ extendable over $S^3$. In this paper we show that the maximum order of extendable cyclic group actions is $4g+4$ when $g$ is even and $4g-4$ when $g$ is odd; the maximum order of extendable abelian group actions is $4g+4$. We also give results of similar questions about extendable group actions over handlebodies.

22pages, 10 figures

References in corpus (1)

• Spin structures and codimension-two homeomorphism extensions

Cited by in corpus (1)

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