Classification of group
actions on surfaces is a question arising from the Nielsen realization problem.
Nielsen gives a classification theorem for cyclic group actions on a closed,
oriented, connected surface which shows that the actions can be classified
by their fixed point data. This paper considers certain group actions on
noncompact, oriented, connected surfaces. The main difficulties are that a
noncompact surface may have infinite genus, and the branch set of an action on a
noncompact surface could be infinite. We introduce the end data and the type of acluster end, and provide a complete classification of cyclic group actions on
noncompact surfaces in terms of fixed point data, end data, and cluster end
types.
