I wrote my Thesis at Valencia University Spain (1982) ,its title was:
Finite type Riemann Surfaces associatted to Schwarz Functions Triangles.
Superficies de Riemann de tipo finito asociadas a las Funciones Triangulares de Schwarz
1.-Alfonso García Marcos Thesis Summary
2.-Alfonso García Marcos Thesis Chapter 2.4
1.-Alfonso García Marcos Thesis Summary
Klein in its The Icosahedron (Dover 1956,p.56) give us a Albebraic Curve
x=x(n)
for the Octahedro (432) also for Tethaedro and Icosahedro..
In order to obtain it , He works with different Invariants Forms.
And in its Vorlesungen Uber die Hipergeometriche funktion (1933, p.229) make this for (5/2,3,2).
In my work I tried to classify all (Platonic Solid,Kepler-Poinsot Solids and in General Schwarz Triangles) of these Algebraic Curves.
In my notation
z=z(w).
I define a Riemann Surface associate to each Regular Tesselation. And after that for Schwarz Triangles.
.
I know Puiseux series in Branch Points.of
1) Proyection z --------> w
2) " w --------> z
and Total Order of Branch.(thanks to Hypergeometric Functions)
Mixing around this concept I can obtain Algebraic Genus and Geometric Density (for overlapping Tesselation) formulas. For cases of Finite Regular Spherical Tesselation are the same than Coxeter Formulas. (Page.32 ,3.1.1,1 and Page 38, 3.1.2,2)
For Schwarz Triangles genus´s is new ( Page 68, 4.3,1) ( Page 69,4.5,1)
Also I , and for Finite Regular Spherical Tesselation I work with regular nets on a Riemann Surface and I obtain with this methode genus same formula.
Klein (1933,p.294 ) also speak about The Transformation Invariant Group of its x=x(n).
In my notation this group is The Automorfism Group of the Covering You can see in my p.34.
I can see (Thanks to Modular Function Landa) Poincare Group of all Riemann Surfaces associatted to Schwarz Triangles as Normal Finite index Subgroups of Γ( 2) ,
Cocients Groups are Geometric Rotation Groups of Tesselations.
I calculate signature of all of these Fuchsian Groups.(Page 71, 4.6)
Then I can say
There are only nine Different Riemann Surface associatted to Schwarz Triangles. More dihedrical cases.
2.-Alfonso García Marcos Thesis Chapter 2.4
Definition of a Riemann Surface Ғ(ξ , θ) associatted to a Schwarz triangle of type (π/ξ , π/ θ ,π/2)
If the Schwarz triangle is like (π/ξ , π/ θ ,π/ ς ) , then the way to define Riemann Surface Ғ(ξ , θ, ς ) is similar that we are going to explain to (π/ξ , π/ θ ,π/2).
From our previous discussions ( Chapters before in our Thesis than 2.4) we know of a function
w=w(z)
which uniquely conformally maps the upper semiplane U {z, Iz>0} onto the spherical rectangular triangle ABC which is the base of the tessellation { ξ , θ }
Triangle {π/ξ , C ;π/2 , A ;π/ θ, B }in w-plane <----------------------------- Upper Semiplane in z-plane
w=w(z)
By Schwarz´s symmetry principle this function can be extended to the lower semiplane I{z, Iz<0} in three differents ways through the segments [-∞, 0] , [0,1] and [1, ∞] producing the functions
w=w1 (z) , w=w2 (z) , w=w3 (z)
which map the lower semiplane I {z, Iz<0} onto the triangles
AB'C , ABC' , BCA' respectively .
We can say that a side of any of the triangles which now appear in this construction is :
· Type 1 if its vertices come from ∞ and 0
· Type 2 if its vertices come from 0 and 1
· Type 3 if its vertices come from ∞ and 1
The process is repeated by prolonging the functions obtained to the upper semiplane U {z, Iz>0} in three ways , thereby obtaining the functions
{ w=wij (z)} 1≤i≤3, 1≤j≤3
where we indicate the function which comes from the analytical extension of w=wi (z) along a line j as wij (z).
The region of definition of w0(z) is U,which defines the element (w0,U).
The region of definition of wi can be considered in two ways ; either as I alone , which gives rise to the element (wi,I) , or as the set S - { 0 ,1, ∞} where S=Sphere understanding that wi acts on U the same as w0 does , and gives rise to the elements (wi, S - { 0 ,1, ∞}).
As for wij(i≠j), only U alone can be considered , thus defining (wij,U) , or rather all S - { 0 ,1, ∞} , understanding that it acts on I as wi does , giving rise to the elements (wij, S - { 0 ,1, ∞}).
In the case of wii we consider it as acting only in S - { 0 ,1, ∞} as does wi , and in U as w0 does , giving rise to the element (wii, S - { 0 ,1, ∞}).
By analytically extending wij along a line k (≠j) we obtain wijk , which defines the elements (wijk,I) in a natural way , and (wijk, S - { 0 ,1, ∞} ) understanding that the latter acts in U as in wij .As before we shall only define the element (wijj , S - { 0 ,1, ∞} ) acts in I as wi and in U as wij.
The process is an iterative one and gives rise to the set of functions elements:
(w0, U)
(wi1i2……i2n, U) ______________________________(i2n ≠ i2n-1, n € N )
(wi1i2……i2n+1, I) ____________________________(i2n+1 ≠ i2n, n € N )
(wi1i2……im, , S - { 0 ,1, ∞} ) ____________________(m € N )
which defines a general analytical function as Ahlfors does (1971 Spanish Version , p.266)
Let us follow two exercises to show the method which allows us to construct a chain which "links" two arbitrary function elements :
1)To link (w12, S - { 0 ,1, ∞} ) with (w31, S - { 0 ,1, ∞} ).
(w12, S - { 0 ,1, ∞}) → (w12,U) → (w122, S - { 0 ,1, ∞}) → (w1,I) → (w11, S - { 0 ,1, ∞}) →
(w0,U) → (w3, S - { 0 ,1, ∞}) → (w31,U) → (w31, S - { 0 ,1, ∞})
2)To link (w1231, U ) with (w21, S - { 0 ,1, ∞} ).
(w1231,U) → (w12311 , S - { 0 ,1, ∞}) → (w123, I) → (w1233, S - { 0 ,1, ∞} ) →
(w12,U) → (w122, S - { 0 ,1, ∞}) → (w1,I) → (w11, S - { 0 ,1, ∞}) →
(w0,U) → (w2, S - { 0 ,1, ∞}) → (w21,U) → (w21, S - { 0 ,1, ∞})