This article is meant to function an advent to the geometry of the motion of discrete teams of Mobius differences. the subject material has now been studied with altering issues of emphasis for over 100 years, the newest advancements being attached with the speculation of 3-manifolds: see, for instance, the papers of Poincare  and Thurston . approximately 1940, the now recognized (but nearly unobtainable) Fenchel-Nielsen manuscript seemed. unfortunately, the manuscript by no means seemed in print, and this extra modest textual content makes an attempt to demonstrate at the least many of the appealing geo metrical rules to be present in that manuscript, in addition to a few newer fabric. The textual content has been written with the conviction that geometrical explana tions are crucial for a whole figuring out of the fabric and that despite the fact that basic a matrix evidence may appear, a geometrical evidence is nearly definitely extra ecocnomic. additional, anywhere attainable, effects may be said in a kind that's invariant less than conjugation, hence making the intrinsic nature of the end result extra obvious. although the subject material is anxious with teams of isometries of hyperbolic geometry, many guides depend on Euclidean estimates and geometry. despite the fact that, the new advancements have back emphasised the necessity for hyperbolic geometry, and i've integrated a finished bankruptcy on analytical (not axiomatic) hyperbolic geometry. it really is was hoping that this bankruptcy will function a "dictionary" offormulae in airplane hyperbolic geometry and as such can be of curiosity and use in its personal correct.
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