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

V