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 with ideal vertices, i.e. where all three angles are strictly positive, the elementary approach of Caratheodory (1954) will be followed. For triangles with one or two ideal vertices, i.e. one or two angles equal to zero, 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, 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 ideal vertices.

Triangles with no ideal vertices

Triangles with one or two ideal vertices

Ideal triangles

Notes

References

• Beardon, Alan F. (1983), The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, ISBN 0-387-90788-2
• Berger, Marcel (2010), Geometry revealed. A Jacob's ladder to modern higher geometry, translated by Lester Senechal, Springer, ISBN 978-3-540-70996-1
• Busemann, Herbert (1955), The geometry of geodesics, Academic Press
• Caratheodory, C. (1954), Theory of functions of a complex variable. Vol. 2., translated by F. Steinhardt., Chelsea Publishing Company
• Davis, Michael W. (2008), The geometry and topology of Coxeter groups, London Mathematical Society Monographs, vol. 32, Princeton University Press, ISBN 978-0-691-13138-2
• de Rham, G. (1971), "Sur les polygones générateurs de groupes fuchsiens", Enseignement Math., 17: 49–61
• Evans, Ronald (1973), "A fundamental region for Hecke's modular group", J. Number Theory, 5: 108–115
• Ford, Lester R. (1951), Automorphic Functions, American Mathematical Society, ISBN 0821837419 , reprint of 1929 edition
• Hardy, G. H.; Wright, E. M. (1979), An introduction to the theory of numbers (Fifth ed.), ISBN 0-19-853170-2 {{citation}}: Text "Oxford University Press" ignored (help)
• Hecke, E. (1935), "Über die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung", Mathematische Annalen (in German), 112: 664–699
• Helgason, Sigurdur (2000), Groups and geometric analysis. Integral geometry, invariant differential operators, and spherical functions, Mathematical Surveys and Monographs, vol. 83, American Mathematical Society, ISBN 0-8218-2673-5
• Hille, Einar (1976), Ordinary differential equations in the complex ___domain, Wiley-Interscience
• Ince, E. L. (1944), Ordinary Differential Equations, Dover Publications
• Magnus, Wilhelm (1974), Noneuclidean tesselations and their groups, Pure and Applied Mathematics, vol. 61, Academic Press
• Maskit, Bernard (1971), "On Poincaré's theorem for fundamental polygons", Advances in Math., 7: 219–230
• Mumford, David; Series, Caroline; Wright, David (2015), Indra's pearls. The vision of Felix Klein, Cambridge University Press, ISBN 978-1-107-56474-9
• Nehari, Zeev (1975), Conformal mapping, Dover Publications
• Series, Caroline (1985), "The modular surface and continued fractions", J. London Math. Soc., 31: 69–80
• Series, Caroline (2015), Continued fractions and hyperbolic geometry, Loughborough LMS Summer School (PDF), retrieved 15 February 2017
• Siegel, C. L. (1971), Topics in complex function theory, Vol. II. Automorphic functions and abelian integrals, translated by A. Shenitzer; M. Tretkoff, Wiley-Interscience, ISBN 0-471-60843-2
• Thurston, William P. (1997), Silvio Levy (ed.), Three-dimensional geometry and topology. Vol. 1., Princeton Mathematical Series, vol. 35, Princeton University Press, ISBN 0-691-08304-5
• Wolf, Joseph A. (2011), Spaces of constant curvature (Sixth ed.), AMS Chelsea Publishing, ISBN 978-0-8218-5282-8