Riemann Surface - Isometries of Riemann Surfaces

Isometries of Riemann Surfaces

The isometry group of a uniformized Riemann surface (equivalently, the conformal automorphism group) reflects its geometry:

• genus 0 – the isometry group of the sphere is the Möbius group of projective transforms of the complex line,
• the isometry group of the plane is the subgroup fixing infinity, and of the punctured plane is the subgroup leaving invariant the set containing only infinity and zero: either fixing them both, or interchanging them (1/z).
• the isometry group of the upper half-plane is the real Möbius group; this is conjugate to the automorphism group of the disk.
• genus 1 – the isometry group of a torus is in general translations (as an Abelian variety), though the square lattice and hexagonal lattice have addition symmetries from rotation by 90° and 60°.
• For genus ≥ 2, the isometry group is finite, and has order at most by Hurwitz's automorphisms theorem; surfaces that realize this bound are called Hurwitz surfaces.
• It's known that every finite group can be realized as the full group of isometries of some riemann surface.
  • For genus 2 the order is maximized by the Bolza surface, with order 48.
  • For genus 3 the order is maximized by the Klein quartic, with order 168; this is the first Hurwitz surface, and its automorphism group is isomorphic to the unique simple group of order 168, which is the second-smallest non-abelian simple group. This group is isomorphic to both PSL(2,7) and PSL(3,2).
  • For genus 4, Bring's surface is a highly symmetric surface.
  • For genus 7 the order is maximized by the Macbeath surface, with order 504; this is the second Hurwitz surface, and its automorphism group is isomorphic to PSL(2,8), the fourth-smallest non-abelian simple group.