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Table of Contents
Intro; Contents; Coxeter Groups and the Davis Complex; 1 Introduction; 2 Group Presentations and Graphs; 2.1 Group Presentations; 2.1.1 A Constructive Approach; 2.2 Some Basic Graph Theory; 2.3 Cayley Graphs for Finitely Presented Groups; 3 Coxeter Groups; 3.1 The Presentation of a Coxeter Group; 3.2 Coxeter Groups and Geometry; 3.2.1 Euclidean Space and Reflections; 3.2.2 Spherical Geometry and Reflections; 3.2.3 Hyperbolic Geometry and Reflections; 3.2.4 The Poincaré Disk Model for Hyperbolic Space; 4 Group Actions on Complexes; 4.1 CW-Complexes; 4.2 Group Actions on CW-Complexes
5 The Cellular Actions of Coxeter Groups: The Davis Complex5.1 Spherical Subsets and the Strict Fundamental Domain; 5.1.1 Spherical Subsets; 5.1.2 The Strict Fundamental Domain; 5.2 The Davis Complex; 5.3 The Mirror Cellulation of Σ; 5.4 The Coxeter Cellulation; 5.4.1 Euclidean Representations; 5.4.2 The Coxeter Cell of Type T; 6 Closing Remarks and Suggested Projects; References; A Tale of Two Symmetries: Embeddable and Non-embeddable Group Actions on Surfaces; 1 Introduction; 2 Determining the Existence of a Group Action; 2.1 Realizing A4 as a Group of Rotations; 2.2 Preliminary Examples
2.3 Signatures2.4 Generating Vectors and Riemann's Existence Theorem; 3 Actions of the Alternating Group A4; 3.1 Signatures for A4-Actions; 4 Embeddable A4-Actions; 4.1 Necessary and Sufficient Conditions for Embeddability of A4; 5 Suggested Projects; References; Tile Invariants for Tackling Tiling Questions; 1 Prologue; 2 Tiling Basics; 3 Tile Invariants; 3.1 Coloring Invariants; 3.2 Boundary Word Invariants; 3.3 Invariants from Local Connectivity; 3.4 The Tile Counting Group; 4 Tile Invariants and Tileability; 5 Enumeration; 6 Concluding Remarks; References
Forbidden Minors: Finding the Finite Few1 Introduction; 2 Properties with Known Kuratowski Set; 3 Strongly Almostâ#x80;#x93;Planar Graphs; 4 Additional Project Ideas; References; Introduction to Competitive Graph Coloring; 1 Introduction; 1.1 Trees and Forests; 1.2 The (r,d)-Relaxed Coloring Game; 1.3 Edge Coloring and Total Coloring; 2 Classifying Forests by Game Chromatic Number; 2.1 Forests with Game Chromatic Number 2; 2.2 Smallest Tree with Game Chromatic Number 4; 3 Relaxed-Coloring Games; 4 The Clique-Relaxed Game; 5 Edge Coloring; 6 Total Coloring; 7 Conclusions and Problems to Consider
5 The Cellular Actions of Coxeter Groups: The Davis Complex5.1 Spherical Subsets and the Strict Fundamental Domain; 5.1.1 Spherical Subsets; 5.1.2 The Strict Fundamental Domain; 5.2 The Davis Complex; 5.3 The Mirror Cellulation of Σ; 5.4 The Coxeter Cellulation; 5.4.1 Euclidean Representations; 5.4.2 The Coxeter Cell of Type T; 6 Closing Remarks and Suggested Projects; References; A Tale of Two Symmetries: Embeddable and Non-embeddable Group Actions on Surfaces; 1 Introduction; 2 Determining the Existence of a Group Action; 2.1 Realizing A4 as a Group of Rotations; 2.2 Preliminary Examples
2.3 Signatures2.4 Generating Vectors and Riemann's Existence Theorem; 3 Actions of the Alternating Group A4; 3.1 Signatures for A4-Actions; 4 Embeddable A4-Actions; 4.1 Necessary and Sufficient Conditions for Embeddability of A4; 5 Suggested Projects; References; Tile Invariants for Tackling Tiling Questions; 1 Prologue; 2 Tiling Basics; 3 Tile Invariants; 3.1 Coloring Invariants; 3.2 Boundary Word Invariants; 3.3 Invariants from Local Connectivity; 3.4 The Tile Counting Group; 4 Tile Invariants and Tileability; 5 Enumeration; 6 Concluding Remarks; References
Forbidden Minors: Finding the Finite Few1 Introduction; 2 Properties with Known Kuratowski Set; 3 Strongly Almostâ#x80;#x93;Planar Graphs; 4 Additional Project Ideas; References; Introduction to Competitive Graph Coloring; 1 Introduction; 1.1 Trees and Forests; 1.2 The (r,d)-Relaxed Coloring Game; 1.3 Edge Coloring and Total Coloring; 2 Classifying Forests by Game Chromatic Number; 2.1 Forests with Game Chromatic Number 2; 2.2 Smallest Tree with Game Chromatic Number 4; 3 Relaxed-Coloring Games; 4 The Clique-Relaxed Game; 5 Edge Coloring; 6 Total Coloring; 7 Conclusions and Problems to Consider