Bordered surfaces in the 3-sphere with maximum symmetry

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 m_α 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.