Bordered surfaces in the 3-sphere with maximum symmetry

Abstract

We consider orientation-preserving actions of finite groups G on pairs (S3,Σ), where Σ denotes a compact connected surface embedded in S3. In a previous paper, we considered the case of closed, necessarily orientable surfaces, determined for each genus g>1 the maximum order of such a G for all embeddings of a surface of genus g, and classified the corresponding embeddings.In the present paper we obtain analogous results for the case of bordered surfaces Σ (i.e. with non-empty boundary, orientable or not). Now the genus g gets replaced by the algebraic genus α of Σ (the rank of its free fundamental group); for each α>1 we determine the maximum order mα of an action of G, classify the topological types of the corresponding surfaces (topological genus, number of boundary components, orientability) and their embeddings into S3. For example, the maximal possibility 12(α−1) is obtained for the finitely many values α=2,3,4,5,9,11,25,97,121 and 241.