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This monograph is Part 1 of a book project intended to give a full account of Jorgensen's theory of punctured torus Kleinian groups and its generalization, with application to knot theory. Although Jorgensen's original work was not published in complete form, it has been a source of inspiration. In particular, it has motivated and guided Thurston's revolutionary study of low-dimensional geometric topology. In this monograph, we give 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.

Kleinian groups. --- Torus (Geometry) --- Knot theory. --- Tore (Géométrie) --- Théorie des noeuds --- Torus (Geometry). --- Kleinian groups --- Knot theory --- Geometry --- Calculus --- Mathematics --- Physical Sciences & Mathematics --- Knots (Topology) --- Anchor ring --- Ring, Anchor --- Groups, Kleinian --- Mathematics. --- Group theory. --- Functions of complex variables. --- Manifolds (Mathematics). --- Complex manifolds. --- Manifolds and Cell Complexes (incl. Diff.Topology). --- Functions of a Complex Variable. --- Group Theory and Generalizations. --- Analytic spaces --- Manifolds (Mathematics) --- Geometry, Differential --- Topology --- Complex variables --- Elliptic functions --- Functions of real variables --- Groups, Theory of --- Substitutions (Mathematics) --- Algebra --- Math --- Science --- Low-dimensional topology --- Surfaces --- Topological spaces --- Discontinuous groups --- Cell aggregation --- Aggregation, Cell --- Cell patterning --- Cell interaction --- Microbial aggregation