Contents
I Introduction 1
1 Introduction 3
1.1 Overview Intersecting Axes 5
1.2 Overview of the intertwining cases 8
1.3 Why an algorithm is needed 10
2 The Acut e Triangle Theorem 13
2.1 Nielsen equivalence 13
2.1.1 Groups and Triangles 14
2.1.2 The Geometric Description of Trace Minimizing . 15
2.1.3 Groups and pentagons 18
2.2 Idea of proof: Acute triangle theorem 19
2.3 Labeling Conventions 20
2.4 Ascending order conventions 21
2.5 The Triangle Algorithm 21
2.6 Q and the last triangle along A 25
2.7 Combining triangle algorithm steps 28
2.8 The sides and heights converge to 0 31
2.9 Acute triangle theorem: proof 31
3 Discreteness Theorem Proof Outline 33
3.1 The Discreteness Theorem 33
3.2 Discreteness theorem 35
3.3 Geometric equivalence theorems 36
II Preliminaries 39
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