Punctured Torus Groups and 2-Bridge Knot Groups (I)

Hirotaka Akiyoshi, Makoto Sakuma, Masaaki Wada, Yasushi Yamashita (auth.)9783540718062, 3540718060

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.

Table of contents :
Front Matter….Pages I-XLIII
Jorgensen’s picture of quasifuchsian punctured torus groups….Pages 1-14
Fricke surfaces and PSL (2, ℂ)-representations….Pages 15-35
Labeled representations and associated complexes….Pages 37-47
Chain rule and side parameter….Pages 49-99
Special examples….Pages 101-132
Reformulation of Main Theorem 1.3.5 and outline of the proof….Pages 133-154
Openness….Pages 155-169
Closedness….Pages 171-214
Algebraic roots and geometric roots….Pages 215-231
Back Matter….Pages 233-256