## Maximum orders of cyclic and abelian extendable actions on surfaces

### Volume 154 / 2018

#### Abstract

A faithful action of a group $G$ on the genus $g \gt 1$ orientable closed surface $\varSigma _g$ is extendable (over the three-dimensional sphere $S^3$), with respect to an embedding $e: \varSigma _g \hookrightarrow S^3$, if there is an action of $G$ on $S^3$ such that $h\circ e=e\circ h$ for any $h \in G$. We show that the maximum order of extendable cyclic group actions on $\varSigma _g$ is $4g+4$ when $g$ is even, and $4g-4$ when $g$ is odd; the maximum order of extendable abelian group actions on $\varSigma _g$ is $4g+4$. We also give the maximum orders of cyclic and abelian group actions on handlebodies.