# 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.

## Group generators

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.

For the torus, the constraint on the free group in two letters is given by $ABA^{-1}B^{-1}=1$ . 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 $ABCA^{-1}B^{-1}C^{-1}=1$ 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 $\Gamma$ in the group $\operatorname {PSL} (2,\mathbb {R} )$ 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:

$A_{1}B_{1}A_{1}^{-1}B_{1}^{-1}A_{2}B_{2}A_{2}^{-1}B_{2}^{-1}\cdots A_{n}B_{n}A_{n}^{-1}B_{n}^{-1}=1$ 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:

$A_{1}A_{1}A_{2}A_{2}\cdots A_{n}A_{n}$ 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

$A_{1}B_{1}A_{1}^{-1}B_{1}^{-1}A_{2}B_{2}A_{2}^{-1}B_{2}^{-1}\cdots A_{n}B_{n}A_{n}^{-1}B_{n}=1$ (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

$A_{1}B_{1}A_{1}^{-1}B_{1}^{-1}A_{2}B_{2}A_{2}^{-1}B_{2}^{-1}\cdots A_{n}B_{n}A_{n}^{-1}B_{n}^{-1}C^{2}=1$ 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

$F=\{z\in \mathbb {H} :d(z,z_{0}) Here, d is a hyperbolic metric on the upper half-plane. The metric fundamental polygon is more usually called the Dirichlet region or the Voronoi polygon.

### 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 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

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 $4\pi (n-1)$ 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.

## 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 $\Gamma$ associated with the standard polygon. For a genus $n$ oriented surface, the group may be given by $2n$ generators $a_{k}$ . These generators are given by the following fractional linear transforms acting on the upper half-plane:

$a_{k}=\left({\begin{matrix}\cos k\alpha &-\sin k\alpha \\\sin k\alpha &\cos k\alpha \end{matrix}}\right)\left({\begin{matrix}e^{p}&0\\0&e^{-p}\end{matrix}}\right)\left({\begin{matrix}\cos k\alpha &\sin k\alpha \\-\sin k\alpha &\cos k\alpha \end{matrix}}\right)$ $\alpha ={\frac {\pi }{4n}}\left(2n-1\right)$ and

$\beta ={\frac {\pi }{4n}}$ and

$p=\ln {\frac {\cos \beta +{\sqrt {\cos 2\beta }}}{\sin \beta }}$ It may be verified that these generators obey the constraint

$a_{0}a_{1}\cdots a_{2n-1}a_{0}^{-1}a_{1}^{-1}\cdots a_{2n-1}^{-1}=1$ 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.