Fundamental polygon

In mathematics, a fundamental polygon can be defined for every compact Riemann surface of genus greater than 0.

In the second case of genus one, the surface is conformally equivalent to a torus C/Λ for some lattice Λ in C. The fundamental polygon of Λ, if assumed convex, may be taken to be either a period parallelogram or a centrally symmetric hexagon, a result first proved by Fedorov in 1891.

In the last case of genus g > 1, the Riemann surface is conformally equivalent to H/Γ where Γ is a Fuchsian group of Möbius transformations.

Using the theory of quasiconformal mappings and the Beltrami equation, it can be shown there is a canonical convex fundamental polygon with 4g sides, first defined by Fricke, which corresponds to the standard presentation of Γ as the group with 2g generators a1, b1, a2, b2, ..., ag, bg and the single relation [a1,b1][a2,b2] ⋅⋅⋅ [ag,bg] = 1, where [a,b] = a b a−1b−1.

Any Riemannian metric on an oriented closed 2-manifold M defines a complex structure on M, making M a compact Riemann surface.

If A is a point of the hexagon, then the lattice is generated by the displacement vectors AB and AC where B and C are the two vertices which are not neighbours of A and not opposite A.

Indeed, the second picture shows how the hexagon is equivalent to the parallelogram obtained by displacing the two triangles chopped off by the segments AB and AC.

If the centre of the hexagon is 0 and the vertices in order are a, b, c, −a, −b and −c, then Λ is the Abelian group with generators a + b and b + c. There are exactly four topologies that can be created by identifying the sides of a rhombus in different ways.

[4] There are several proofs of this, some of the more recent ones related to results in convexity theory, the geometry of numbers and circle packing, such as the Brunn–Minkowski inequality.

[6][7] Coxeter's proof proceeds by assuming that there is a centrally symmetric convex polygon C with 2m sides.

Let v, e and f be the number of vertices, edges and faces in this tiling (taking into account identifications in the quotient space).

It is not possible for both αi and βi to be even, since otherwise ± 1/2 xi would be a point of Λ on a side, which contradicts C being a fundamental domain.

This is an example of a Dirichlet domain or Voronoi diagram: since complex translations form an Abelian group, so commute with the action of Λ, these concepts coincide.

The canonical fundamental domain for Λ = Z + Zω with Im ω > 0 is either a symmetric convex parallelogram or hexagon with centre 0.

Thus the classification of compact Riemann surfaces can be reduced to the study of possible groups Γ.

The genus is zero if the covering space is the Riemann sphere; one if it is the complex plane; and greater than one if it is the unit disk or upper halfplane.

The group Γ can thus be identified with a lattice Λ in C and X with a quotient C/Λ, as described in the section on fundamental polygons in genus one.

As Poincaré observed, each such polygon has special properties, namely it is convex and has a natural pairing between its sides.

Without assumptions on the convexity of the polygon, complete proofs have been given by Maskit and de Rham, based on an idea of Siegel, and can be found in Beardon (1983), Iversen (1992) and Stillwell (1992).

When all the angles equal π/2g, this establishes the tiling by regular 4g-sided hyperbolic polygons and hence the existence of a particular compact Riemann surface of genus g as a quotient space.

This construction shows that the classification of closed orientable 2-manifolds up to diffeomorphism or homeomorphism can be reduced to the case of compact Riemann surfaces.

[10] The classification up to homeomorphism and diffeomorphism of compact Riemann surfaces can be accomplished using the fundamental polygon.

Then following Nevanlinna and Jost, the fundamental domain can be modified in steps to yield a non-convex polygon with vertices lying in a single orbit of Γ and piecewise geodesic sides.

In order the labelling is so that Γ is generated by the ai and bi subject to the single relation Using the theory of intersection numbers, it follows that the shape obtained by joining vertices by geodesics is also a proper polygon, not necessarily convex, and is also a fundamental domain with the same group elements giving the pairing.

Let F(t) be the solution of the Beltrami equation tμ normalised to fix three points on the unit circle.

[16] Assuming without loss of generality that 0 lies in the interior of a convex fundamental polygon C and g is an element of Γ, the ray from 0 to g(0)—the hyperbolic geodesic—passes through a succession of translates of C. Each of these is obtained from the previous one by applying a generator of Γ or a fixed product of generators (if successive translates meet in a vertex).

This treatment is an analytic counterpart of the classical topological classification of orientable 2-dimensional polyhedra presented in Seifert & Threlfall (1934).

Using the theory of quasiconformal mappings of Ahlfors and Bers, Keen (1965) gave a new, shorter and more precise version of Fricke's construction.

, with exactly one defining constraint, The genus of the Riemann surface H/Γ is g. The area of the standard fundamental polygon is

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.