Editing Riemann surface (section)

== Isometries of Riemann surfaces ==
The [[isometry group]] of a uniformized Riemann surface (equivalently, the conformal [[Automorphism#Automorphism_group|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&nbsp;(1/''z'').
* the isometry group of the [[Poincaré half-plane model|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 generated by translations (as an [[Abelian variety]]) and the rotation by 180°. In special cases there can be additional rotations and reflections.<ref name="car20"/>
* For genus {{nowrap|''g'' ≥ 2}}, the isometry group is finite, and has order at most {{nowrap|84(''g'' − 1)}}, by [[Hurwitz's automorphisms theorem]]; surfaces that realize this bound are called '''Hurwitz surfaces'''.
* It is known that every finite group can be realized as the full group of isometries of some Riemann surface.<ref>{{cite book |first=L. |last=Greenberg |chapter=Maximal groups and signatures |title=Discontinuous Groups and Riemann Surfaces: Proceedings of the 1973 Conference at the University of Maryland |series=Ann. Math. Studies |volume=79 |year=1974 |pages=207–226 |isbn=0691081387 |chapter-url=https://books.google.com/books?id=oriXEV6VoM0C&pg=PA207 }}</ref>
** 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 {{nowrap|[[PSL(2, 7)]]}} and {{nowrap|[[PSL(2,7)|PSL(3, 2)]]}}.
** For genus 4, [[Bring's curve|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 {{nowrap|PSL(2, 8)}}, the fourth-smallest non-abelian simple group.