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Bibliographic Details
Main Authors:	Ceccherini-Silberstein, Tullio 1966- (Author), Coornaert, Michel (Author)
Format:	Book
Language:	English
Published:	Berlin [u.a.] Springer 2010
Series:	Springer Monographs in Mathematics
Subjects:	Zellularer Automat Gruppentheorie
Links:	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=021128952&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
Physical Description:	XIX, 439 S.
ISBN:	9783642140334

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Record in the Search Index

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adam_text	Contents 1 Cellulair Automata ........................................ 1 1.1 The Configuration Set and the Shift Action ................ 1 1.2 The Prodiscrete Topology ............................... 3 1.3 Periodic Configurations ................................. 3 1.4 Cellular Automata ...................................... 6 1.5 Minimal Memory ....................................... 14 1.6 Cellular Automata over Quotient Groups .................. 15 1.7 Induction and Restriction of Cellular Automata ............ 16 1.8 Cellular Automata with Finite Alphabets .................. 20 1.9 The Prodiscrete Uniform Structure ....................... 22 1.10 Invertitile Cellular Automata ............................. 24 Notes ...................................................... 27 Exercises .................................................. 29 2 Residually Finite Groups ................................. 37 2.1 Definition and First Examples ........................... 37 2.2 Stability Properties of Residually Finite Groups ............ 40 2.3 Residual Finiteness of Free Groups ....................... 42 2.4 Hopfian Groups ........................................ 44 2.5 Automorphism Groups of Residually Finite Groups ......... 45 2.6 Examples of Finitely Generated Groups Which Are Not Residually Finite ....................................... 47 2.7 Dynamical Characterization of Residual Rniteness .......... 50 Notes ...................................................... 51 Exercises .................................................. 52 3 Surjunctive Groups ....................................... 57 3.1 Definition ............................................. 57 3.2 Stability Properties of Surjunctive Groups ................. 58 3.3 Surjunctivity of Locally Residually Finite Groups ........... 59 xiv Contents 3.4 Marked Groups ........................................ 61 3.5 Expansive Actions on Uniform Spaces ..................... 64 3.6 Gromov s Injectivity Lemma ............................. 65 3.7 Closedness of Marked Surjunctive Groups ................. 67 Notes ...................................................... 68 Exercises .................................................. 68 4 Amenable Groups ........................................ 77 4.1 Measures and Means .................................... 77 4.2 Properties of the Set of Means ........................... 82 4.3 Measures and Means on Groups .......................... 83 4.4 Definition of Amenability ................................ 85 4.5 Stability Properties of Amenable Groups .................. 88 4.6 Solvable Groups ........................................ 92 4.7 The F0lner Conditions .................................. 94 4.8 Paradoxical Decompositions ............................. 98 4.9 The Theorems of Tarski and F0lner ....................... 99 4.10 The Fixed Point Property ...............................103 Notes ......................................................105 Exercises ..................................................106 5 The Garden of Eden Theorem ............................ Ill 5.1 Garden of Eden Configurations and Garden of Eden Patterns 111 5.2 Pre-injective Maps ......................................112 5.3 Statement of the Garden of Eden Theorem ................114 5.4 Interiors, Closures, and Boundaries .......................115 5.5 Mutually Erasable Patterns ..............................121 5.6 Tilings ................................................122 5.7 Entropy ...............................................125 5.8 Proof of the Garden of Eden Theorem ....................128 5.9 Surjunctivity of Locally Residually Amenable Groups .......131 5.10 A Surjective but Not Pre-injective Cellular Automaton over F2 ................................................133 5.11 A Pre-injective but Not Surjective Cellular Automaton over F2 ................................................133 5.12 A Characterization of Amenability in Terms of Cellular Automata .............................................135 5.13 Garden of Eden Patterns for Life .........................136 Notes ......................................................138 Exercises ..................................................139 6 Finitely Generated Amenable Groups ....................151 6.1 The Word Metric .......................................151 6.2 Labeled Graphs ........................................153 6.3 Cayley Graphs .........................................156 6.4 Growth Functions and Growth Types .....................160 Contents xv 6.5 The Growth Rate ......................................168 6.6 Growth of Subgroups and Quotients ......................170 6.7 A Finitely Generated Metabelian Group with Exponential Growth ...............................................173 6.8 Growth of Finitely Generated Nilpotent Groups ............175 6.9 The Grigorchuk Group and Its Growth ....................178 6.10 The F0lner Condition for Finitely Generated Groups ........191 6.11 Amenability of Groups of Subexponential Growth ..........192 6.12 The Theorems of Kesten and Day ........................193 6.13 Quasi-Isometries .......................................204 Notes ......................................................214 Exercises ..................................................217 7 Local Embeddability and Sofie Groups ....................233 7.1 Local Embeddability ....................................234 7.2 Local Embeddability and Ultraproducts ...................243 7.3 LEF-Groups and LEA-Groups ...........................246 7.4 The Hamming Metric ...................................251 7.5 Sofie Groups ...........................................254 7.6 Sofie Groups and Metric Ultraproducts of Finite Symmetric Groups ................................................260 7.7 A Characterization of Finitely Generated Sofie Groups ......265 7.8 Surjunctivity of Sofie Groups ............................272 Notes ......................................................275 Exercises ..................................................278 8 Linear Cellular Automata ................................283 8.1 The Algebra of Linear Cellular Automata .................284 8.2 Configurations with Finite Support .......................288 8.3 Restriction and Induction of Linear Cellular Automata ......289 8.4 Group Rings and Group Algebras ........................291 8.5 Group Ring Representation of Linear Cellular Automata .... 294 8.6 Modules over a Group Ring ..............................299 8.7 Matrix Representation of Linear Cellular Automata .........301 8.8 The Closed Image Property ..............................305 8.9 The Garden of Eden Theorem for Linear Cellular Automata . 308 8.10 Pre-injective but not Surjective Linear Cellular Automata . . . 314 8.11 Surjective but not Pre-injective Linear Cellular Automata . . . 315 8.12 Invertible Linear Cellular Automata ......................317 8.13 Pre-injectivity and Surjectivity of the Discrete Laplacian .... 321 8.14 Linear Surjunctivity ....................................324 8.15 Stable Finiteness of Group Algebras ......................327 8.16 Zero-Divisors in Group Algebras and Pre-injectivity of One-Dimensional Linear Cellular Automata .............330 Notes ......................................................335 Exercises ..................................................338 xvl Contents A Nets and the Tychonoff Product Theorem ................343 A.I Directed Sets ..........................................343 A. 2 Nets in Topological Spaces ...............................343 A.3 Initial Topology ........................................346 A. 4 Product Topology ......................................346 A. 5 The Tychonoff Product Theorem .........................347 Notes ......................................................349 В Uniform Structures .......................................351 B.I Uniform Spaces ........................................351 B.2 Uniformly Continuous Maps .............................353 B.3 Product of Uniform Spaces ..............................355 B.4 The Hausdorff-Bourbaki Uniform Structure on Subsets ......356 Notes ......................................................358 С Symmetric Groups .......................................359 C.I The Symmetric Group ..................................359 C.2 Permutations with Finite Support ........................360 C.3 Conjugacy Classes in Symo(X) ...........................362 C.4 The Alternating Group ..................................363 D Free Groups ..............................................367 D.I Concatenation of Words .................................367 D.2 Definition and Construction of Free Groups ................367 D.3 Reduced Forms ........................................373 D.4 Presentations of Groups .................................375 D.5 The Klein Ping-Pong Theorem ...........................376 E Inductive Limits and Projective Limits of Groups ........379 E.I Inductive Limits of Groups ..............................379 E.2 Projective Limits of Groups ..............................380 F The Banach-Alaoglu Theorem ............................383 F.I Topological Vector Spaces ...............................383 F.2 The Weak-* Topology ...................................384 F.3 The Banach-Alaoglu Theorem ............................384 G The Markov-Kakutani Fixed Point Theorem ..............387 G.I Statement of the Theorem ...............................387 G.2 Proof of the Theorem ...................................387 Notes ......................................................389 H The Hall Harem Theorem ................................391 H.I Bipartite Graphs .......................................391 H.2 Matchings .............................................393 Contents xvii H. 3 The Hall Marriage Theorem .............................394 H. 4 The Hall Harem Theorem ...............................399 Notes ......................................................401 I Complements of Functional Analysis ......................403 1.1 The Baire Theorem .....................................403 1.2 The Open Mapping Theorem ............................404 1.3 Spectra of Linear Maps .................................406 1.4 Uniform Convexity .....................................407 J Ultrafilters ............................................... 409 J.I Filters and Ultrafilters .................................. 409 J. 2 Limits Along Filters .................................... 412 Notes ...................................................... 415 Open Problems ...............................................417 Comments .................................................418 References ....................................................421 List of Symbols ...............................................429 Index .........................................................433
any_adam_object	1
author	Ceccherini-Silberstein, Tullio 1966- Coornaert, Michel
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dewey-tens	510 - Mathematics
discipline	Informatik Mathematik
format	Book
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spellingShingle	Ceccherini-Silberstein, Tullio 1966- Coornaert, Michel Cellular Automata and Groups Zellularer Automat (DE-588)4190671-8 gnd Gruppentheorie (DE-588)4072157-7 gnd
subject_GND	(DE-588)4190671-8 (DE-588)4072157-7
title	Cellular Automata and Groups
title_auth	Cellular Automata and Groups
title_exact_search	Cellular Automata and Groups
title_full	Cellular Automata and Groups Tullio Ceccherini-Silberstein; Michel Coornaert
title_fullStr	Cellular Automata and Groups Tullio Ceccherini-Silberstein; Michel Coornaert
title_full_unstemmed	Cellular Automata and Groups Tullio Ceccherini-Silberstein; Michel Coornaert
title_short	Cellular Automata and Groups
title_sort	cellular automata and groups
topic	Zellularer Automat (DE-588)4190671-8 gnd Gruppentheorie (DE-588)4072157-7 gnd
topic_facet	Zellularer Automat Gruppentheorie
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=021128952&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT ceccherinisilbersteintullio cellularautomataandgroups AT coornaertmichel cellularautomataandgroups

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