Classifying Finite Group Actions on Surfaces of Low Genus

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The problem of classifying all finite group actions, up to topological equivalence, on a surface of low genus is considered. Several new examples of construction and classification of actions are given. A program for enumerating all finite group actions on a surface of low genus is outlined and the complete classification is worked out for genus 2 and 3. As a by-product all conjugacy classes of finite subgroups of mapping class groups for genus 2 and 3 are determined.