Maximum Orders of Cyclic and Abelian Extendable Actions on Surfaces

TL;DR

This paper determines the maximum possible orders of cyclic and abelian groups acting extendably on surfaces embedded in 3-spheres, providing exact bounds based on the genus of the surface.

Contribution

It establishes precise maximum orders for extendable cyclic and abelian group actions on surfaces in $S^3$, and explores similar bounds for handlebodies.

Findings

Maximum order of extendable cyclic actions is 4g+4 for even g and 4g-4 for odd g.

Maximum order of extendable abelian actions is 4g+4.

Results extend to group actions on handlebodies.

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.