The classical free motion of a mass point on a compact surface of constant negative curvature is the most chaotic possible. The quantal description of this motion is under development.
In this review we (a) bring together the mathematical tools needed for the problem; (b) survey the classical results; (c) develop the quantal description; (d) discuss the relations between the classical and quantal results; (e) offer some conjectures towards the diagnosis of chaoticity in quantum systems.
The text is divided into chapters and sections; each section ends with a brief summary which also serves as a local abstract. Chapter I gives the motivations. Chapter II describes various models of the pseudosphere, or Bolyai-Lobachevsky plane. Chapters HI and IV analyse the classical motions constrained to surfaces of constant negative curvature, and the salient results on ergodicity, mixing, etc.... Chapters V and VI specify the quantum problem and exhibit the relevant mathematical tools. Chapter VII exhibits the remarkable exact connections between the classical and quantal results which are embodied in Selberg's trace formula or are consequences of it. Chapter VIII deals with the semiclassical connections between the two descriptions, including the quantal equivalents of such classical notions as the Liapunov exponent and the Poincare section map.
Chapter IX contains some numerical results obtained by C. Schmit on the spectrum of a simple case, in particular on its fluctuations and their interpretation. Chapter X summarises our conclusions.
The Appendices from A to N are integral parts of the review and are only separated from the text in order not to interrupt the flow. They contain details which clarify mathematical or conceptual points. Similarly, the figures should also be used to complement and clarify the text.