The Poincaré Upper Half-Plane

In Chap. 2, we considered a model for elliptic geometry, in which any two geodesics intersect, so that there are no parallels. Now we want to investigate a model for hyperbolic geometry, in which there are infinitely many geodesics through a given point which are parallel to a given geodesic. This sort of geometry was discovered by Bolyai, Gauss, and Lobatchevsky in the 1820s. However, Gauss never published his results, perhaps because the idea was controversial. In fact, Gauss embittered Bolyai by claiming precedence in a letter to Bolyai’s father (see Gauss [199, Vol. 8, pp. 220–225]). The subject of non-Euclidean geometry was still controversial when Lewis Carroll “repudiated hyperbolic geometry in 1888 as being too fanciful” (see the article of Coxeter in COSRIMS [110, p. 55]). Models for hyperbolic geometry were obtained first by Liouville, then by Beltrami in 1868, and by Klein in 1870. Poincaré rediscovered the Liouville–Beltrami upper half-plane model in 1882 and this space is usually called the Poincaré upper half-plane, though some call it the Lobatchevsky upper half-plane (but see Milnor [469]). Poincaré [517] also considered discontinuous groups of transformations of the hyperbolic upper half-plane as well as the functions left invariant by these groups and we intend to do the same. The geometric foundations for such work were laid by Gauss in 1827 (see Gauss [199, Vol. 4, pp. 217–258]) and by Riemann in 1854 (see Riemann [542, pp. 272–287]). These important papers of Gauss and Riemann are discussed from a modern perspective in Spivak [615, Vol. II].