Fundamental polygon

In mathematics, each closed surface in the sense of geometric topology can be constructed from an even-sided oriented polygon, called a fundamental polygon, by pairwise identification of its edges.

This construction can be represented as a string of length 2n of n distinct symbols where each symbol appears twice with exponent either +1 or -1. The exponent -1 signifies that the corresponding edge has the orientation opposing the one of the fundamental polygon.

Contents

1 Examples

2 Standard fundamental polygons

3 Fundamental polygon of a compact Riemann surface

3.1 Metric fundamental polygon
3.2 Standard fundamental polygon
3.3 Example
3.4 Area

4 References

Examples

sphere: $A A - 1$
projective plane: $A A$
Klein bottle: $A B A - 1 B$ or $A A B B$
torus: $A B A - 1 B - 1$ or $A B C A - 1 B - 1 C - 1$

Standard fundamental polygons

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

$A_1 B_1 A_1^{-1} B_1^{-1}A_2 B_2 A_2^{-1} B_2^{-1}\cdots A_n B_n A_n^{-1} B_n^{-1}$

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

$A_1 A_1 A_2 A_2 \cdots A_n A_n$

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

$F=\{z \in \mathbb{H} : d(z,z_0) < d(z,gz_0) \;\; \forall g\in \Gamma \}$

Here, d is a hyperbolic metric on the upper half-plane.

This fundamental polygon is a fundamental domain.
This fundamental polygon is convex in that the geodesic joining any two points of the polygon is contained entirely inside the polygon.
The diameter of F is less than or equal to the diameter of H/Γ. In particular, the closure of F is compact.
If Γ has no fixed points in H and H/Γ is compact, then F will have finitely many sides.
Each side of the polygon is a geodesic arc.
For every side s of the polygon, there is precisely one other side s' such that gs=s' for some g in Γ. Thus, this polygon will have an even number of sides.
The set of group elements g that join sides to each other are generators of Γ, and there is no smaller set that will generate Γ.
The upper half-plane is tiled by the closure of F under the action of Γ. That is, $H=\cup_{g\in\Gamma}\, g\overline{F}$ where $\overline{F}$ is the closure of F.

Standard fundamental polygon

Given any metric fundamental polygon F, one can construct, with a finite number of steps, another fundamental polygon, the standard fundamental polygon, which has an additional set of noteworthy properties:

The vertices of the standard polygon are all equivalent. By vertex is meant the point where two sides meet. By equivalent, it is meant that each vertex can be carried to any of the other vertices by some g in Γ.
The number of sides is divisible by four.
A given element g of Γ will carry at most one side of the polygon to another. Thus, the sides can be marked off in pairs. Since the action of Γ is orientation-preserving, if one side is called $A$ , then the other of the pair can be marked with the opposite orientation $A - 1$ .
The edges of the standard polygon can be arranged so that the list of adjacent sides takes the form $A_1 B_1 A_1^{-1} B_1^{-1}A_2 B_2 A_2^{-1} B_2^{-1}\cdots A_n B_n A_n^{-1} B_n^{-1}$ . That is, pairs of sides can be arranged so that they interleave in this way.
The standard polygon is convex.
The sides can be arranged to be geodesic arcs.

The above construction is sufficient to guarantee that each side of the polygon is a closed (non-trivial) loop in the manifold H/Γ. As such, each side can thus an element of the fundamental group $\pi_1 (\mathbb{H}/\Gamma)$ . In particular, the fundamental group $\pi_1 (\mathbb{H}/\Gamma)$ has 2n generators $A_1, B_1, A_2, B_2, \cdots A_n B_n$ , with exactly one defining constraint,

$A_1 B_1 A_1^{-1} B_1^{-1}A_2 B_2 A_2^{-1} B_2^{-1}\cdots A_n B_n A_n^{-1} B_n^{-1}=1$ .

The genus of the resulting manifold H/Γ is n.

Example

Note that the metric fundamental polygon and the standard fundamental polygon will usually have a different number of sides. Thus, for example, the standard fundamental polygon on a torus is a fundamental parallelogram. By contrast, the metric fundamental polygon is six-sided, a hexagon. This can be most easily seen by noting that the sides of the hexagon are perpendicular bisectors of the edges of the parallelogram. That is, one picks a point in the lattice, and then considers the set of straight lines joining this point to nearby neighbors. Bisecting each such line by another perpendicular line, the smallest space walled off by this second set of lines is a hexagon.

In fact, this last construction works in generality: picking a point x, one then considers the geodesics between x and gx for g in Γ. Bisecting these geodesics is another set of curves, the locus of points equidistant between x and gx. The smallest region enclosed by this second set of lines is the metric fundamental polygon.

Area

The area of the standard fundamental polygon is $4π(n - 1)$ where n is the genus of the Riemann surface (equivalently, where 4n 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.

References

Hershel M. Farkas and Irwin Kra, Riemann Surfaces (1980), Springer-Verlag, New York. ISBN 0-387-90465-4.
Jurgen Jost, Compact Riemann Surfaces (2002), Springer-Verlag, New York. ISBN 3-540-43299-X.