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Topics-in-combinatorial-group-theory

Group Theory

"... in a series of extraordinarily influential papers between 1910 and 1914, Max Dehn proposed and partly solved a number of problems about finitely-presented groups, thereby heralding in the birth of a new subject, combinatorial group theory. Thus the subject became endowed and encumbered by many of the problems that had stimulated its birth. The problems were generally concerned with various classes of groups and were of the following kind: Are all the groups in a given class finite? Finitely-generated? Finitely presented? What are the conjugates of a given element in a given group? What are the subgroups of that group? Is there an algorithm for deciding for every pair of groups in a given class whether they are isomorphic? And so on. The object of combinatorial group theory is the systematic development of algebraic techniques to settle such questions. In view of the scope of the subject and the extraordinary variety of groups involved, it is not surprising that no really general theory exists. However much has been accomplished and a wide variety of techniques and methods has been developed with wide application and potential."

Chapter 1 HistoryEdit

Chapter 2 The Weak Burnside ProblemEdit

Chapter 3 Free groups, the calculus of presentations and...Edit

Chapter 4 Recursively presented groups, word problems and...Edit

Chapter 5 Affine algebraic sets and the representative theory of finitely generated groupsEdit

Section 5.1. BackgroundEdit

Section 5.2. Some basic algebraic geometryEdit

Section 5.3. More basic algebraic geometryEdit

Section 5.4. Useful notions from topologyEdit

Section 5.5. MorphismsEdit

Section 5.6. DimensionEdit

Section 5.7. Representations of the free group of rank two in SL(2,C)Edit

  • Lemma 16: The logic here is out-of-order. Start by taking a representation of F (the free group on two generators) rather than H, then note that by Exercises 2(3)(iv) on page 98 this representation factors through H. Then the rest is fine.
  • Lemma 18: After the line $ B^2 + ({\rm tr} B) B + I = 0 $, multiply both sides by $ B^{-1} $ and rearrange.

Section 5.8. Affine algebraic sets of charactersEdit

Chapter 6 Generalized free products and HNN extensionsEdit

Chapter 7 Groups acting on treesEdit