Fundamental polygon

Examples

Group generators

For the set of standard, symmetric shapes, the symbols of the edges of the polygon may be understood to be the generators of a group. Then, the polygon, written in terms of group elements, becomes a constraint on the free group generated by the edges, giving a group presentation with one constraint.

Thus, for example, given the Euclidean plane R 2 , let the group element A act on the plane as A ( x , y ) = ( x + 1 , y ) while B ( x , y ) = ( x , y + 1 ) . Then A , B generate the lattice Γ = Z 2 , and the torus is given by the quotient space (a homogeneous space) T = R 2 / Z 2 . More generally, the two generators A , B can be taken to generate a parallelogram tiling, of fundamental parallelograms.

For the torus, the constraint on the free group in two letters is given by A B A − 1 B − 1 = 1 . This constraint is trivially embodied in the action on the plane given above. Alternately, the plane can be tiled by hexagons, and the centers of the hexagons form a hexagonal lattice. Identifying opposite edges of the hexagon again leads to the torus, this time, with the constraint A B C A − 1 B − 1 C − 1 = 1 describing the action of the hexagonal lattice generators on the plane.

In practice, most of the interesting cases are surfaces with negative curvature, and are thus realized by a discrete lattice Γ in the group PSL ⁡ ( 2 , R ) acting on the upper half-plane. Such lattices are known as Fuchsian groups.

Standard fundamental polygons

An orientable closed surface of genus n has the following standard fundamental polygon:

This fundamental polygon can be viewed as the result of gluing n tori together, and hence the surface is sometimes called the n-fold torus. ("Gluing" two surfaces means cutting a disk out of each and identifying the circular boundaries of the resulting holes.)

A non-orientable closed surface of (non-orientable) genus n has the following standard fundamental polygon:

Alternately, the non-orientable surfaces can be given in one of two forms, as n Klein bottles glued together (this may be called the n-fold Klein bottle, with non-orientable genus 2n), or as n glued real projective planes (the n-fold crosscap, with non-orientable genus n). The n-fold Klein bottle is given by the 4n-sided polygon

(note the final B n is missing the superscript −1; this flip, as compared to the orientable case, being the source of the non-orientability). The (2n + 1)-fold crosscap is given by the 4n+2-sided polygon

That these two cases exhaust all the possibilities for a compact non-orientable surface was shown by Henri Poincaré.

Fundamental polygon of a compact Riemann surface

The fundamental polygon of a (hyperbolic) compact Riemann surface has a number of important properties that relate the surface to its Fuchsian model. That is, a hyperbolic compact Riemann surface has the upper half-plane as the universal cover, and can be represented as a quotient manifold H/Γ where Γ is a non-Abelian group isomorphic to the deck transformation group of the surface. The cosets of the quotient space have the standard fundamental polygon as a representative element. In the following, note that all Riemann surfaces are orientable.

Metric fundamental polygon

Given a point z 0 in the upper half-plane H, and a discrete subgroup Γ of PSL(2,R) that acts freely discontinuously on the upper half-plane, then one can define the metric fundamental polygon as the set of points

Here, d is a hyperbolic metric on the upper half-plane. The metric fundamental polygon is more usually called the Dirichlet polygon .

Fricke canonical polygon

Given a Riemann surface of genus g greater than one, Fricke described another fundamental polygon, the Fricke canonical polygon, which is a very special example of a Dirichlet polygon. The polygon is related to the standard presentation of the fundamental group of the surface. Fricke's original construction is complicated and described in Fricke & Klein (1897). Using the theory of quasiconformal mappings of Ahlfors and Bers, Keen (1965) gave a new, shorter and more precise version of Fricke's construction. The Fricke canonical polygon has the following properties:

The above construction is sufficient to guarantee that each side of the polygon is a closed (non-trivial) loop in the Riemann surface H/Γ. As such, each side can thus an element of the fundamental group π 1 ( H / Γ ) ≡ Γ . In particular, the fundamental group π 1 ( H / Γ ) has 2g generators A 1 , B 1 , A 2 , B 2 , ⋯ A g , B g , with exactly one defining constraint,

The genus of the Riemann surface H/Γ is g.

Area

The area of the standard fundamental polygon is 4 π ( g − 1 ) where g is the genus of the Riemann surface (equivalently, where 4g is the number of the sides of the polygon). Since the standard polygon is a representative of H/Γ, the total area of the Riemann surface is equal to the area of the standard polygon. The area formula follows from the Gauss–Bonnet theorem and is in a certain sense generalized through the Riemann–Hurwitz formula.

Explicit form for standard polygons

Explicit expressions can be given for the regular standard 4g-sided polygon, with rotational symmetry. In this case, that of a genus g Riemann surface with g-fold rotational symmetry, the group may be given by 2 g generators a k . These generators are given by the following fractional linear transforms acting on the upper half-plane:

for 0 ≤ k < 2 g . The parameters are given by

and

and

It may be verified that these generators obey the constraint

which gives the totality of the group presentation.

Generalizations

In higher dimensions, the idea of the fundamental polygon is captured in the articulation of homogeneous spaces.