1. Jorgensen's picture of quasifuchsian punctured torus
groups ....................................................... 1
1.1. Punctured torus groups, Ford domains and
EPH-decompositions ...................................... 2
1.2. Jorgensen's theorem for quasifuchsian punctured torus
groups (I) .............................................. 7
1.3. Jorgensen's theorem for quasifuchsian punctured torus
groups (II) ............................................ 12
1.4. The topological ideal polyhedral complex Trg(ν) dual
to Spine(ν) ............................................ 13
2. Fricke surfaces and PSL(2,C)-representations ................ 15
2.1. Fricke surfaces and their fundamental groups ........... 16
2.2. Type-preserving representations ........................ 21
2.3. Markoff maps and type-preserving representations ....... 26
2.4. Markoff maps and complex probability maps .............. 29
2.5. Miscellaneous properties of discrete groups ............ 33
3. Labeled representations and associated complexes ............ 37
3.1. The complex  (ρ,σ) and upward Markoff maps ............ 38
3.2. The complexes (ρ,Σ) and (Σ) ......................... 41
3.3. Labeled representation ρ = (ρ,ν) and the complexes
(ρ) and (ν) ......................................... 44
3.4. Virtual Ford domain .................................... 44
4. Chain rule and side parameter ............................... 49
4.1. Chain rule for isometric circles ....................... 50
4.2. Side parameter ......................................... 56
4.3. ∈-terminal triangles .................................. 66
4.4. Basic properties of ∈-terminal triangles .............. 70
4.5. Relation between side parameters at adjacent
triangles .............................................. 78
4.6. Transition of terminal triangles ....................... 82
4.7. Proof of Lemma 4.5.5. .................................. 89
4.8. Representations which are weakly simple at σ ........... 95
5. Special examples ........................................... 101
5.1. Real representations .................................. 102
5.2. Isosceles representations and thin labels ............. 106
5.3. Groups generated by two parabolic transformations ..... 117
5.4. Imaginary representations ............................. 126
5.5. Representations with accidental parabolic/elliptic
transformations ....................................... 127
6. Reformulation of Main Theorem 1.3.5 and outline
of the proof ............................................... 133
6.1. Reformulation of Main Theorem 1.3.5. .................. 134
6.2. Route map of the proof of Modified Main
Theorem 6.1.11 ........................................ 136
6.3. The cellular structure of ∂Eh(ρ) ...................... 138
6.4. Applying Poincare's theorem on fundamental
polyhedra ............................................. 142
6.5. Proof of Theorem 6.1.8 (Good implies quasifuchsian) ... 144
6.6. Structure of the complex ΔE and the proof of
Theorem 6.1.12. ....................................... 147
6.7. Characterization of Σ(ν) for good labeled
representations ....................................... 151
7. Openness ................................................... 155
7.1. Hidden isometric hemispheres .......................... 155
7.2. Proof of Proposition 6.2.1 (Openness) -
Thick Case - .......................................... 159
7.3. Proof of Proposition 6.2.1 (Openness) - Thin Case - ... 165
8. Closedness ................................................. 171
8.1. Proof of Proposition 6.2.3 (SameStratum) .............. 172
8.2. Proof of Proposition 6.2.7 (Convergence) .............. 178
8.3. Route map of the proof of Proposition 6.2.4
(Closedness) .......................................... 180
8.4. Reduction of Proposition 8.3.5 - The condition
HausdorffConvergence - ................................ 182
8.5. Classification of simplices of (ν) ................... 184
8.6. Proof of Proposition 8.4.4 (F∞(ξ) ⊂ ∂Eh(ρ∞,L0)) ....... 185
8.7. Accidental parabolic transformation ................... 187
8.8. Proof of Proposition 8.4.5 - length 1 case - .......... 189
8.9. Proof of Proposition 8.4.5 - length ≥ 2 case -
(Step 1) .............................................. 191
8.10.Proof of Proposition 8.4.5 - length ≥ 2 case -
(Step 2) .............................................. 203
8.11.Proof of Proposition 8.4.5 - length ≥ 2 case -
(Step 3) .............................................. 206
8.12.Proof of Proposition 8.3.6 ............................ 209
9. Algebraic roots and geometric roots......................... 215
9.1. Algebraic roots ....................................... 215
9.2. Unique existence of the geometric root ................ 227
9.3. Continuity of roots and continuity of intersections ... 229
A. Appendix ................................................... 233
A.l. Basic facts concerning the Ford domain ................ 233
References .................................................... 239
Notation ...................................................... 245
Index ......................................................... 249
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