Here is the first part of a work that provides a full account of
Jorgensen's theory of punctured torus Kleinian groups and its
generalization. It offers an elementary and self-contained
description of Jorgensen's theory with a complete proof. Through
various informative illustrations, readers are naturally led to an
intuitive, synthetic grasp of the theory, which clarifies how a
very simple fuchsian group evolves into complicated Kleinian
groups.
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