Profinite graphs and groups

This book offers a detailed introduction to graph theoretic methods in profinite groups and applications to abstract groups. It is the first to provide a comprehensive treatment of the subject. The author begins by carefully developing relevant notions in topology, profinite groups and homology, including free products of profinite groups, cohomological methods in profinite groups, and fixed points of automorphisms of free pro-p groups. The final part of the book is dedicated to applications of the profinite theory to abstract groups, with sections on finitely generated subgroups of free groups, separability conditions in free and amalgamated products, and algorithms in free groups and finite monoids. Profinite Graphs and Groups will appeal to students and researchers interested in profinite groups, geometric group theory, graphs and connections with the theory of formal languages. A complete reference on the subject, the book includes historical and bibliographical notes as well as a discussion of open questions and suggestions for further reading. Read more... Abstract: This book offers a detailed introduction to graph theoretic methods in profinite groups and applications to abstract groups. It is the first to provide a comprehensive treatment of the subject. The author begins by carefully developing relevant notions in topology, profinite groups and homology, including free products of profinite groups, cohomological methods in profinite groups, and fixed points of automorphisms of free pro-p groups. The final part of the book is dedicated to applications of the profinite theory to abstract groups, with sections on finitely generated subgroups of free groups, separability conditions in free and amalgamated products, and algorithms in free groups and finite monoids. Profinite Graphs and Groups will appeal to students and researchers interested in profinite groups, geometric group theory, graphs and connections with the theory of formal languages. A complete reference on the subject, the book includes historical and bibliographical notes as well as a discussion of open questions and suggestions for further reading Profinite Graphs and Groups, doi:10.1007/978-3-319-61199-0 Profinite Graphs and Groups 3 Preface 6 Chapter 1: Preliminaries 14 1.1 Inverse Limits 15 1.2 Profinite Spaces 16 1.3 Profinite Groups 17 Pseudovarieties C 18 Generators 19 G-Spaces and Continuous Sections 19 Order of a Profinite Group and Sylow Subgroups 20 1.4 Pro-C Topologies in Abstract Groups 21 1.5 Free Groups 22 1.6 Free and Amalgamated Products of Groups 23 1.7 Profinite Rings and Modules 24 Exact Sequences 26 The Functors Hom(-,-) 26 Projective and Injective Modules 27 1.8 The Complete Group Algebra 28 G-Modules 28 Complete Tensor Products 29 1.9 The Functors ExtLambdan (-,-) and TornLambda(-,-) 30 The Functors ExtLambdan (-,-) 30 The Functors TornLambda(-,-) 32 1.10 Homology and Cohomology of Profinite Groups 33 Cohomology of Profinite Groups 33 Special Maps in Cohomology 36 Homology of Profinite Groups 37 Duality Homology-Cohomology 37 (Co)induced Modules and Shapiro's Lemma 38 1.11 (Co)homological Dimension 38 Part I: Basic Theory 40 Chapter 2: Profinite Graphs 42 2.1 First Notions and Examples 42 2.2 Groups Acting on Profinite Graphs 54 2.3 The Chain Complex of a Graph 58 2.4 pi-Trees and C-Trees 61 2.5 Cayley Graphs and C-Trees 70 Chapter 3: The Fundamental Group of a Profinite Graph 75 3.1 Galois Coverings 75 3.2 G(Gamma|Delta) as a Subgroup of Aut(Gamma) 84 3.3 Universal Galois Coverings and Fundamental Groups 86 3.4 0-Transversals and 0-Sections 89 3.5 Existence of Universal Coverings 94 3.6 Subgroups of Fundamental Groups of Graphs 101 3.7 Universal Coverings and Simple Connectivity 103 3.8 Fundamental Groups and Projective Groups 107 3.9 Fundamental Groups of Quotient Graphs 108 3.10 pi-Trees and Simple Connectivity 112 3.11 Free Pro-C Groups and Cayley Graphs 117 3.12 Change of Pseudovariety 119 Chapter 4: Profinite Groups Acting on C-Trees 122 4.1 Fixed Points 122 4.2 Faithful and Irreducible Actions 130 Chapter 5: Free Products of Pro-C Groups 148 5.1 Free Pro-C Products: The External Viewpoint 148 5.2 Subgroups Continuously Indexed by a Space 156 5.3 Free Pro-C Products: The Internal Viewpoint 159 5.4 Profinite G-Spaces vs the Weight w(G) of G 164 5.5 Basic Properties of Free Pro-C Products 168 5.6 Free Products and Change of Pseudovariety 175 5.7 Constant and Pseudoconstant Sheaves 178 Chapter 6: Graphs of Pro-C Groups 188 6.1 Graphs of Pro-C Groups and Specializations 188 6.2 The Fundamental Group of a Graph of Pro-C Groups 191 Uniqueness of the Fundamental Group 196 6.3 The Standard Graph of a Graph of Pro-C Groups 204 6.4 Injective Graphs of Pro-C Groups 216 6.5 Abstract vs Profinite Graphs of Groups 218 6.6 Action of a Pro-C Group on a Profinite Graph with Finite Quotient 224 6.7 Notes, Comments and Further Reading: Part I 227 Abstract Graph of Finite Groups (G, Gamma) over an Infinite Graph Gamma 229 Part II: Applications to Profinite Groups 231 Chapter 7: Subgroups of Fundamental Groups of Graphs of Groups 233 7.1 Subgroups 233 7.2 Normal Subgroups 237 7.3 The Kurosh Theorem for Free Pro-C Products 242 Chapter 8: Minimal Subtrees 246 8.1 Minimal Subtrees: The Abstract Case 247 8.2 Minimal Subtrees: Abstract vs Profinite Trees 250 Trees Associated with Virtually Free Groups 251 8.3 Graphs of Residually Finite Groups and the Tits Line 254 8.4 Graph of a Free Product of Groups and the Tits Line 259 Chapter 9: Homology and Graphs of Pro-C Groups 266 9.1 Direct Sums of Modules and Homology 266 9.2 Corestriction and Continuously Indexed Families of Subgroups 268 9.3 The Homology Sequence of the Action on a Tree 274 9.4 Mayer-Vietoris Sequences 276 9.5 Homological Characterization of Free Pro-p Products 279 9.6 Pro-p Groups Acting on C-Trees and the Kurosh Theorem 281 Chapter 10: The Virtual Cohomological Dimension of Profinite Groups 288 10.1 Tensor Product of Complexes 288 10.2 Tensor Product Induction for a Complex 290 10.3 The Torsion-Free Case 299 10.4 Groups Virtually of Finite Cohomological Dimension: Periodicity 300 10.5 The Torsion Case 304 10.6 Pro-p Groups with a Free Subgroup of Index p 318 10.7 Counter Kurosh 321 10.8 Fixed Points of Automorphisms of Free Pro-p Groups 326 10.9 Notes, Comments and Further Reading: Part II 331 M. Hall Pro-p Groups 332 Part III: Applications to Abstract Groups 335 Chapter 11: Separability Conditions in Free and Polycyclic Groups 337 11.1 Separability Conditions in Abstract Groups 337 11.2 Subgroup Separability in Free-by-Finite Groups 341 11.3 Products of Subgroups in Free Abstract Groups 345 11.4 Separability Properties of Polycyclic Groups 350 Chapter 12: Algorithms in Abstract Free Groups and Monoids 356 12.1 Algorithms for Subgroups of Finite Index 356 12.2 Closure of Finitely Generated Subgroups in Abstract Free Groups 360 12.3 Algorithms for Monoids 366 The Kernel of a Finite Monoid 371 The Mal'cev Product of Pseudovarieties of Monoids 373 Chapter 13: Abstract Groups vs Their Profinite Completions 375 13.1 Free-by-Finite Groups vs Their Profinite Completions 375 13.2 Polycyclic-by-Finite Groups vs Their Profinite Completions 385 Chapter 14: Conjugacy in Free Products and in Free-by-Finite Groups 389 14.1 Conjugacy Separability in Free-by-Finite Groups 389 14.2 Conjugacy Subgroup Separability in Free-by-Finite Groups 392 14.3 Conjugacy Distinguishedness in Free-by-Finite Groups 395 Chapter 15: Conjugacy Separability in Amalgamated Products 397 15.1 Abstract Free Products with Cyclic Amalgamation 398 15.2 Normalizers in Amalgamated Products of Groups 402 15.3 Conjugacy Separability of Amalgamated Products 405 15.4 Amalgamated Products, Quasi-potency and Subgroup Separability 411 15.5 Amalgamated Products and Products of Cyclic Subgroups 413 15.6 Amalgamated Products and Normalizers of Cyclic Subgroups 417 15.7 Amalgamated Products and Intersections of Cyclic Subgroups 418 15.8 Amalgamated Products and Conjugacy Distinguishedness 421 15.9 Conjugacy Separability of Certain Iterated Amalgamated Products 424 15.10 Examples of Conjugacy Separable Groups 424 15.11 Notes, Comments and Further Reading: Part III 428 Subgroup Separability and Free Products 429 Conjugacy Separability, Subgroups and Extensions 433 Conjugacy Distinguished Subgroups 433 Appendix A: Abstract Graphs 434 A.1 The Fundamental Group of an Abstract Graph 434 The Star of a Vertex 435 Paths 435 A.2 Coverings of Abstract Graphs 440 A.3 Foldings 446 A.4 Algorithms 447 Intersection of Finitely Generated Subgroups 448 A.5 Notes, Comments and Further Reading 450 Appendix B: Rational Sets in Free Groups and Automata 451 B.1 Finite State Automata: Review and Notation 451 B.2 The Classical Function rho 452 B.3 Rational Subsets in Free Groups 453 B.4 Notes, Comments and Further Reading 455 References 456 Index of Symbols 463 Index of Authors 465 Index of Terms 467 This Book Offers A Detailed Introduction To Graph Theoretic Methods In Profinite Groups And Applications To Abstract Groups. It Is The First To Provide A Comprehensive Treatment Of The Subject. The Author Begins By Carefully Developing Relevant Notions In Topology, Profinite Groups And Homology, Including Free Products Of Profinite Groups, Cohomological Methods In Profinite Groups, And Fixed Points Of Automorphisms Of Free Pro-p Groups. The Final Part Of The Book Is Dedicated To Applications Of The Profinite Theory To Abstract Groups, With Sections On Finitely Generated Subgroups Of Free Groups, Separability Conditions In Free And Amalgamated Products, And Algorithms In Free Groups And Finite Monoids. Profinite Graphs And Groups Will Appeal To Students And Researchers Interested In Profinite Groups, Geometric Group Theory, Graphs And Connections With The Theory Of Formal Languages. A Complete Reference On The Subject, The Book Includes Historical And Bibliographical Notes As Well As A Discussion Of Open Questions And Suggestions For Further Reading. 1 Preliminaries -- Part I Basic Theory -- 2 Profinite Graphs -- 3 The Fundamental Group Of A Profinite Graph -- 4 Profinite Groups Acting On C-trees -- 5 Free Products Of Pro-c Groups -- 6 Graphs Of Pro-c Groups -- Part Ii Applications To Profinite Groups -- 7 Subgroups Of Fundamental Groups Of Graphs Of Groups -- 8 Minimal Subtrees -- 9 Homology And Graphs Of Pro-c Groups -- 10 The Virtual Cohomological Dimension Of Profinite Groups -- Part Iii Applications To Abstract Groups -- 11 Separability Conditions In Free And Polycyclic Groups -- 12 Algorithms In Free Groups And Monoids -- 13 Abstract Groups Vs Their Profinite Completions -- 14 Conjugacy In Free Products And In Free-by-finite Groups -- 15 Conjugacy Separability In Amalgamated Products -- Appendix A Abstract Graphs -- Appendix B Rational Sets In Free Groups And Automata -- Bibliography -- Index. Luis Ribes. Includes Bibliographical References (pages 453-459) And Indexes. Front Matter ....Pages I-XV Preliminaries (Luis Ribes)....Pages 1-26 Front Matter ....Pages 27-28 Profinite Graphs (Luis Ribes)....Pages 29-61 The Fundamental Group of a Profinite Graph (Luis Ribes)....Pages 63-109 Profinite Groups Acting on \(\mathcal{C}\)-Trees (Luis Ribes)....Pages 111-136 Free Products of Pro-\(\mathcal{C}\) Groups (Luis Ribes)....Pages 137-176 Graphs of Pro-\(\mathcal{C}\) Groups (Luis Ribes)....Pages 177-219 Front Matter ....Pages 221-222 Subgroups of Fundamental Groups of Graphs of Groups (Luis Ribes)....Pages 223-235 Minimal Subtrees (Luis Ribes)....Pages 237-256 Homology and Graphs of Pro-\(\mathcal{C}\) Groups (Luis Ribes)....Pages 257-278 The Virtual Cohomological Dimension of Profinite Groups (Luis Ribes)....Pages 279-325 Front Matter ....Pages 327-328 Separability Conditions in Free and Polycyclic Groups (Luis Ribes)....Pages 329-347 Algorithms in Abstract Free Groups and Monoids (Luis Ribes)....Pages 349-367 Abstract Groups vs Their Profinite Completions (Luis Ribes)....Pages 369-382 Conjugacy in Free Products and in Free-by-Finite Groups (Luis Ribes)....Pages 383-390 Conjugacy Separability in Amalgamated Products (Luis Ribes)....Pages 391-427 Back Matter ....Pages 429-471