Finite group actions on bordered surfaces of small genus

Abstract

This paper considers finite group actions on compact bordered surfaces — quotients of unbordered orientable surfaces under the action of a reflectional symmetry. Classification of such actions (up to topological equivalence) is carried out by means of the theory of non-euclidean crystallographic groups, and determination of normal subgroups of finite index in these groups, up to conjugation within their automorphism group. A result of this investigation is the determination, up to topological equivalence, of all actions of groups of finite order 6 or more on compact (orientable or non-orientable) bordered surfaces of algebraic genus p for 2 ≤ p ≤ 6 . We also study actions of groups of order less than 6, or of prime order, on bordered surfaces of arbitrary algebraic genus p ≥ 2 .