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.
Real projective plane
For the set of standard, symmetrical 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 , let the group element act on the plane as while . Then generate the lattice , and the torus is given by the quotient space (a homogeneous space) . More generally, the two generators 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 . 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 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 in the group 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 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 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
- 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, where 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 , then the other of the pair can be marked with the opposite orientation .
- The edges of the standard polygon can be arranged so that the list of adjacent sides takes the form . 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 . In particular, the fundamental group has 2n generators , with exactly one defining constraint,
The genus of the resulting manifold H/Γ is n.
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.
The area of the standard fundamental polygon is 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.
Explicit form for standard polygons
Explicit expressions can be given for the standard polygons. One of the more useful forms is in terms of the group associated with the standard polygon. For a genus oriented surface, the group may be given by generators . These generators are given by the following fractional linear transforms acting on the upper half-plane:
for . The parameters are given by
It may be verified that these generators obey the constraint
which gives the totality of the group presentation.
In higher dimensions, the idea of the fundamental polygon is captured in the articulation of homogeneous spaces.
- Alan F. Beardon, The Geometry of Discrete Groups (1983), Springer-Verlag, New York. ISBN 0-387-90788-2.
- 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.
- Example of sphere construction from fundamental polygon.