Euclidean symmetry of closed surfaces immersed in 3-space

Abstract Given a finite group G of orientation-preserving isometries of euclidean 3-space E 3 and a closed surface S, an immersion f : S → E 3 is in G-general position if f ( S ) is invariant under G, points of S have disk neighborhoods whose images are in general position, and no singular points of f ( S ) lie on an axis of rotation of G. For such an immersion, there is an induced action of G on S whose Riemann–Hurwitz equation satisfies certain natural restrictions. We classify which restricted Riemann–Hurwitz equations are realized by a G-general position immersion of S. This generalizes work by various authors on euclidean symmetry of closed surfaces embedded in E 3 . The analysis involves a detailed study of immersions of the quotient surface S / G in the orbifold E 3 / G .