Schwarz triangle function

In mathematics, the Schwarz triangle function was introduced by H. A. Schwarz as the inverse function of the conformal mapping uniformizing a Schwarz triangle, i.e. a geodesic triangle in the upper half plane with angles which are either 0 or of the form $π$ over a positive integer greater than one. Applying successive hyperbolic reflections in its sides, such a triangle generates a tessellation of the upper half plane (or the unit disk after composition with the Cayley transform). The conformal mapping of the upper half plane onto the interior of the geodesic triangle generalizes the Schwarz–Christoffel transformation. Through the theory of the Schwarzian derivative, it can be expressed as the quotient of two solutions of a hypergeometric differential equation with real coefficients and singular points at 0, 1 and ∞. By the Schwarz reflection principle, the discrete group generated by hyperbolic reflections in the sides of the triangle induces an action on the two dimensional space of solutions. On the orientation-preserving normal subgroup, this two dimensional representation corresponds to the monodromy of the ordinary differential equation and induces a group of Möbius transformations on quotients of solutions. Since the triangle function is the inverse function of such a quotient, it is therefore an automorphic function for this discrete group of Möbius transformations. This is a special case of a general method of Henri Poincaré that associates automorphic forms with ordinary differential equations with regular singular points. In the special case of ideal triangles, where all the angles are zero, the tessellation corresponds to the Farey tessellation and the triangle function yields the modular lambda function.

Hyperboloid and Klein models

Convex polygons

Tessellation by Schwarz triangles

In this section tessellations of the hyperbolic upper half plane by Schwarz triangles will be discussed using elementary methods. For triangles without "cusps"—angles equal to zero or equivalently vertices on the real axis—the elementary approach of Caratheodory (1954) will be followed. For triangles with one or two cusps, elementary arguments of Evans (1973), simplifying the approach of Hecke (1935), will be used: in the case of a Schwarz triangle with one angle zero and another a right angle, the orientation-preserving subgroup of the reflection group of the triangle is a Hecke group. For an ideal triangle in which all angles are zero, so that all vertices lie on the real axis, the existence of the tessellation will be established by relating it to the Farey series described in Hardy & Wright (1979) and Series (2015). In this case the tessellation can be considered as that associated with three touching circles on the Riemann sphere, a limiting case of configurations associated with three disjoint non-nested circles and their reflection groups, the so-called "Schottky groups", described in detail in Mumford, Series & Wright (2015). Alternatively—by dividing the ideal triangle into six triangles with angles 0, $π$ /2 and $π$ /3—the tessellation by ideal triangles can be understood in terms of tessellations by triangles with one or two cusps.

Triangles without cusps

Suppose that the hyperbolic triangle Δ has angles $π$ /a, $π$ /b and $π$ /c with a, b, c integers greater than 1. The hyperbolic area of Δ equals $π$ – $π$ /a – $π$ /b – $π$ /c, so that

{1 \over a}+{1 \over b}+{1 \over c}<1.

The construction of a tessellation will first be carried out for the case when a, b and c are greater than 2.^[1]

The original triangle Δ gives a convex polygon P₁ with 3 vertices. At each of the three vertices the triangle can be successively reflected through edges emanating from the vertices to produce 2m copies of the triangle where the angle at the vertex is $π$ /m. The triangles do not overlap except at the edges, half of them have their orientation reversed and they fit together to tile a neighborhood of the point. The union of these new triangles together with the original triangle form a connected shape P₂. It is made up of triangles which only intersect in edges or vertices, forms a convex polygon with all angles less than $π$ and each side being the edge of a reflected triangle. This can be seen more clearly by noting that some triangles or tiles are added twice, the three which have a side in common with the original triangle. The rest have only a vertex in common. A more systematic way of performing the tiling is first to add a tile to each side (the reflection of the triangle in that edge) and then fill in the gaps at each vertex. This results in a total of 3 + (2a – 3) + (2b - 3) + (2c - 3) = 2(a + b + c) - 6 new triangles. The new vertices are of two two types. Those which are vertices of the triangles attached to sides of the original triangle, which are connected to 2 vertices of Δ. Each of these lie in three new triangles which intersect at that vertex. The remainder are connected to a unique vertex of Δ and belong to two new triangles which have a common edge. Thus there are 3 + (2a – 4) + (2b - 4) + (2c - 4) = 2(a + b + c) - 9 new vertices. By construction there is no overlapping. To see that P₂ is convex, it suffices to see that the angle between sides meeting at a new vertex make an angle less than $π$ . But the new vertices lies in two or three new triangles, which meet at that vertex, so the angle at that vertex is less than 2 $π$ /3 or $π$ , as required.

This process can be repeated for P₂ to get P₃ by first adding tiles to each edge of P₂ and then filling in the tiles round each vertex of P₂. Then the process can be repeated from P₃, to get P₄ and so on, successively producing P_n from P_{n – 1}. It can be checked inductively that these are all convex polygons, with non-overlapping tiles. Indeed as in the first step of the process there are two types of tile in building P_n from P_{n – 1}, those attached to an edge of P_{n – 1} and those attached to a single vertex. Similarly there are two types of vertex, one in which two new tiles meet and those in which three tiles meet. So provided that no tiles overlap the previous argument shows that angles at vertices do not exceed $π$ and implies that P_n is a convex polygon.

The equality above for a, b and c implies that if one of the angles is a right angle, say a = 2, then both b and c are greater than 2 and one of them, b say, must be greater than 3. In this case, reflecting the triangle across the side AB gives an isosceles hyperbolic triangle with angles $π$ /c, $π$ /c and 2 $π$ /b. The construction of the tessellation above through increasing convex polygons adapts word for word to this case except that around the vertex with angle 2 $π$ /b, only b—and not 2b—copies of the triangle are required to tile a neighborhood of the vertex. This is possible because the doubled triangle is isosceles.^[2]

Triangles with one or two cusps

Ideal triangles

Notes

^ Caratheodory 1954, pp. 177–181
^ Caratheodory 1954, pp. 181–182

References

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[1] Caratheodory 1954, pp. 177–181

[2] Caratheodory 1954, pp. 181–182

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