Infinite Group Theory: From the Past to the Future

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Contents Introduction Gerhard Rosenberger Dennis Spellman 1. Groups With the Weak Minimal Condition on Non-Permutable Subgroups 1. Introduction 2. Some Results Concerning the Weak Minimal Condition 3. Groups with the Weak Minimal Condition on Non-Permutable Subgroups 4. Main Results References 2. A Survey: Shamir Threshold Scheme and Its Enhancements 1. Introduction 2. Entropy 3. Shamir (t, n) Threshold Scheme 3.1. Access structure 3.2. Perfect and ideal scheme 3.3. Proactive scheme 3.4. Verifiable scheme 3.5. Enhancements by one-way function and RSA 3.5.1. Applying one-way function in Shamir scheme 3.5.2. Using one-way functions and RSA in a Shamir scheme 4. Extension to Any General Access Structure 5. Relation with Reed-Solomon Code 6. Shamir Ramp Scheme 7. Information Disposal Algorithm and Making Secret Short 8. Secure Multiparty Computation 9. Private Information Retrieval and Shamir Scheme 10. Practical Applications 11. Other Platforms 12. Conclusions and Future Research References 3. The Zappa-Szep Product of Left-Orderable Groups 1. Introduction 2. Appendix References 4. Totally Disconnected Groups From Baumslag- Solitar Groups 1. Introduction 2. The Scale Function 3. Baumslag-Solitar Groups 4. Commensurated Subgroups 5. Construction of the Totally Disconnected Group 5.1. An alternative construction of Gm,n 6. Computing Scales for Gm,n 7. The Modular Function, Flat Rank, and Scale-Multiplicative Subsemigroups 8. The Local Structure of Gm,n References 5. Elementary and Universal Theories of Nonabelian Commutative Transitive and CSA Groups 1. A Generalization of Commutative Transitivity 2. Reduced Products 3. Preliminaries from Logic 4. The Elementary Theory Case 5. Algebraic Geometry over Groups and a Theorem of Myasnikov and Remeslennikov 6. Question References 6. Commutative Transitivity and the CSA Property 1. Introduction 2. Basic Material on CT and CSA Groups 3. Equivalences Between CT and CSA and the Groups PSL(2,K) 4. Infinite CT non-CSA Groups References 7. The Universal Theory of Free Burnside Groups of Large Prime Exponent 1. Introduction 2. Preliminaries from Logic 3. Algebraic Geometry Over Groups 4. The Case of Prime Exponent 4.1. The universal theory with respect to the base language L0 4.2. The universal theory with respect to the extended language L0[G] 5. The Case of General Odd Exponent 6. Independence of the Axioms References 8. Primitive Curve Lengths on Pairs of Pants 1. Introduction 2. Preliminaries: Notation and Terminology 3. Preliminaries: Summary of Some Prior Results 4. Preliminaries: Standard Hexagons, Notation and Facts 5. Preliminaries: Essential Self-Intersections and Winding and Unwinding Sequences 6. Length Inequalities References 9. Drawing Inferences Under Maximum Entropy From Relational Probabilistic Knowledge Using Group Theory 1. Introduction 2. Representing Probabilistic Conditional Knowledge 3. Exploiting the Conditional Structure 4. Basics of Elimination Theory 5. Compiling Knowledge Bases 6. Drawing Inferences 7. Future Work 8. Conclusion References 10. On Some Infinite-Dimensional Linear Groups and the Structure of Related Modules References 11. On New Analogs of Some Classical Group Theoretical Results in Lie Rings 1. Introduction 2. Proof of Theorem A 3. Proof of Theorem B 4. Proof of Theorem C 5. Proof of Theorem D References 12. Log-Space Complexity of the Conjugacy Problem in Wreath Products 1. Introduction 2. Preliminaries 2.1. Log-space computability 2.2. Wreath products and representing their elements 2.3. Useful algorithms 2.4. On the conjugacy problem for A B 2.5. On the conjugacy problem in wreath products of abelian groups 2.6. On the conjugacy problem in iterated wreath products References 13. Group Presentations, Cayley Graphs and Markov Processes 1. Introduction 2. Group Presentations and their Cayley Graphs 3. Sabidussi’s Theorem 4. Choosing Random Elements in a Group, Random Walks on the Cayley Graph 5. Markov Processes 6. Markov Processes and Free Groups References