The Automorphism Group on the Riemann Sphere
نویسنده
چکیده
In order to study the geometries of a hyperbolic plane, it is necessary to understand the set of transformations that map from the space to itself. This paper shows that in the Poincaré half-plane model, this group of transformations is the general Möbius group. This is done by showing that the general Möbius group is a group of homeomorphisms that map circles to circles on the extended complex plane Ĉ, and that all homeomorphisms that send circles to circles on Ĉ are Möbius transformations.
منابع مشابه
On Riemann surfaces of genus g with 4g automorphisms
We determine, for all genus g ≥ 2 the Riemann surfaces of genus g with exactly 4g automorphisms. For g 6= 3, 6, 12, 15 or 30, this surfaces form a real Riemann surface Fg in the moduli space Mg: the Riemann sphere with three punctures. We obtain the automorphism groups and extended automorphism groups of the surfaces in the family. Furthermore we determine the topological types of the real form...
متن کاملSpatial statistics for lattice points on the sphere I: Individual results
We study the spatial distribution of point sets on the sphere obtained from the representation of a large integer as a sum of three integer squares. We examine several statistics of these point sets, such as the electrostatic potential, Ripley's function, the variance of the number of points in random spherical caps, and the covering radius. Some of the results are conditional on t...
متن کاملOn Cyclic Groups of Automorphisms of Riemann Surfaces
The question of extendability of the action of a cyclic group of automorphisms of a compact Riemann surface is considered. Particular attention is paid to those cases corresponding to Singerman's list of Fuchsian groups which are not nitely-maximal, and more generally to cases involving a Fuchsian triangle group. The results provide partial answers to the question of which cyclic groups are the...
متن کاملOn the automorphism groups of convex domains in C
We establish that every bounded convex domain in C with an automorphism orbit accumulation at a boundary point at which the domain has a sphere contact from inside admits a non-compact 1-parameter subgroup of automorphisms. Notice that this in particular implies that no Teichmüller domain of a Riemann surface of genus g > 1 can be holomorphically imbedded as a convex domain in C .
متن کاملElliptic Curves
1. Throughout P = C ∪ {∞} denotes the Riemann sphere, H denotes the upper half plane, C∗ denotes the multiplicative group of complex numbers, and P = (C \ {0})/C∗ denotes n dimensional complex projective space. For w ∈ C \{0} let [w] := wC∗ denote the corresponding point of P. For A ∈ GLn+1(C) let MA denote the corresponding automorphism of projective space so that MA([w]) = [Aw]. Identify P an...
متن کامل