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Table of Contents
Intro
Foreword
Preface
Contents
Partial List of Notations
1 Cellular Automata
1.1 Summary
1.1.1 Configuration Spaces
1.1.2 The Prodiscrete Topology
1.1.3 Periodic Configurations
1.1.4 Cellular Automata
1.1.5 The Curtis-Hedlund-Lyndon Theorem
1.1.6 Induction and Restriction of Cellular Automata
1.1.7 Invertible Cellular Automata
1.2 Exercises
2 Residually Finite Groups
2.1 Summary
2.1.1 Equivalent Definitions of Residual Finiteness
2.1.2 The Class of Residually Finite Groups
2.1.3 Divisible Groups
2.1.4 Hopfian Groups
2.2 Exercises
3 Surjunctive Groups
3.1 Summary
3.1.1 Definition of Surjunctivity
3.1.2 The Class of Surjunctive Groups
3.1.3 Expansive Actions
3.1.4 Compactness of the Space of MarkedSurjunctive Groups
3.2 Exercises
4 Amenable Groups
4.1 Summary
4.1.1 Equivalent Definitions of Amenability
4.1.2 The Class of Amenable Groups
4.1.3 Amenability of Solvable and Nilpotent Groups
4.2 Exercises
5 The Garden of Eden Theorem
5.1 Summary
5.1.1 Interiors, Closures, and Boundaries
5.1.2 Tilings
5.1.3 Pre-injective Maps
5.1.4 Garden of Eden Configurations
5.1.5 Entropy
5.1.6 The Garden of Eden Theorem
5.2 Exercises
6 Finitely Generated Groups
6.1 Summary
6.1.1 The Word Metric
6.1.2 Labeled Graphs
6.1.3 Cayley Graphs
6.1.4 Growth of Finitely Generated Groups
6.1.5 The Grigorchuk Group
6.1.6 The Kesten-Day Characterization of Amenability
6.1.7 Quasi-Isometries
6.2 Exercises
7 Local Embeddability and Sofic Groups
7.1 Summary
7.1.1 Local Embeddability
7.1.2 LEF and LEA-Groups
7.1.3 The Hamming Metric
7.1.4 Sofic Groups
7.1.5 Sofic Groups and Ultraproducts
7.1.6 Geometric Characterization of Finitely Generated Sofic Groups
7.1.7 Surjunctivity of Sofic Groups
7.2 Exercises
8 Linear Cellular Automata
8.1 Summary
8.1.1 Rings
8.1.2 Group Rings
8.1.3 NZD-Groups, Unique-Product Groups, and Orderable Groups
8.1.4 Linear Shift Spaces
8.1.5 Linear Cellular Automata
8.1.6 Restriction and Induction of Linear Cellular Automata
8.1.7 Group Ring Representation of Linear Cellular Automata
8.1.8 Matrix Representation of Linear Cellular Automata
8.1.9 The Closed Image Property for LinearCellular Automata
8.1.10 Invertible Linear Cellular Automata
8.1.11 Mean Dimension
8.1.12 The Garden of Eden Theorem for LinearCellular Automata
8.1.13 The Discrete Laplacian
8.1.14 Linear Surjunctivity
8.2 Exercises
References
Index
Foreword
Preface
Contents
Partial List of Notations
1 Cellular Automata
1.1 Summary
1.1.1 Configuration Spaces
1.1.2 The Prodiscrete Topology
1.1.3 Periodic Configurations
1.1.4 Cellular Automata
1.1.5 The Curtis-Hedlund-Lyndon Theorem
1.1.6 Induction and Restriction of Cellular Automata
1.1.7 Invertible Cellular Automata
1.2 Exercises
2 Residually Finite Groups
2.1 Summary
2.1.1 Equivalent Definitions of Residual Finiteness
2.1.2 The Class of Residually Finite Groups
2.1.3 Divisible Groups
2.1.4 Hopfian Groups
2.2 Exercises
3 Surjunctive Groups
3.1 Summary
3.1.1 Definition of Surjunctivity
3.1.2 The Class of Surjunctive Groups
3.1.3 Expansive Actions
3.1.4 Compactness of the Space of MarkedSurjunctive Groups
3.2 Exercises
4 Amenable Groups
4.1 Summary
4.1.1 Equivalent Definitions of Amenability
4.1.2 The Class of Amenable Groups
4.1.3 Amenability of Solvable and Nilpotent Groups
4.2 Exercises
5 The Garden of Eden Theorem
5.1 Summary
5.1.1 Interiors, Closures, and Boundaries
5.1.2 Tilings
5.1.3 Pre-injective Maps
5.1.4 Garden of Eden Configurations
5.1.5 Entropy
5.1.6 The Garden of Eden Theorem
5.2 Exercises
6 Finitely Generated Groups
6.1 Summary
6.1.1 The Word Metric
6.1.2 Labeled Graphs
6.1.3 Cayley Graphs
6.1.4 Growth of Finitely Generated Groups
6.1.5 The Grigorchuk Group
6.1.6 The Kesten-Day Characterization of Amenability
6.1.7 Quasi-Isometries
6.2 Exercises
7 Local Embeddability and Sofic Groups
7.1 Summary
7.1.1 Local Embeddability
7.1.2 LEF and LEA-Groups
7.1.3 The Hamming Metric
7.1.4 Sofic Groups
7.1.5 Sofic Groups and Ultraproducts
7.1.6 Geometric Characterization of Finitely Generated Sofic Groups
7.1.7 Surjunctivity of Sofic Groups
7.2 Exercises
8 Linear Cellular Automata
8.1 Summary
8.1.1 Rings
8.1.2 Group Rings
8.1.3 NZD-Groups, Unique-Product Groups, and Orderable Groups
8.1.4 Linear Shift Spaces
8.1.5 Linear Cellular Automata
8.1.6 Restriction and Induction of Linear Cellular Automata
8.1.7 Group Ring Representation of Linear Cellular Automata
8.1.8 Matrix Representation of Linear Cellular Automata
8.1.9 The Closed Image Property for LinearCellular Automata
8.1.10 Invertible Linear Cellular Automata
8.1.11 Mean Dimension
8.1.12 The Garden of Eden Theorem for LinearCellular Automata
8.1.13 The Discrete Laplacian
8.1.14 Linear Surjunctivity
8.2 Exercises
References
Index