Combinatorial group theory is a loosely defined subject, with close connections to topology and logic. With surprising frequency, problems in a wide variety of disciplines, including differential equations, automorphic functions and geometry, have been distilled into explicit questions about groups, typically of the following kind: Are the groups in a given class finite (e.g., the Burnside problem)? 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 or not? The objective 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. These notes, bridging the very beginning of the theory to new results and developments, are devoted to a number of topics in combinatorial group theory and serve as an introduction to the subject on the graduate level.
I History.- 1. Introduction.- 2. The beginnings.- 3. Finitely presented groups.- 4. More history.- 5. Higman's marvellous theorem.- 6. Varieties of groups.- 7. Small Cancellation Theory.- II The Weak Burnside Problem.- 1. Introduction.- 2. The Grigorchuk-Gupta-Sidki groups.- 3. An application to associative algebras.- III Free groups, the calculus of presentations and the method of Reidemeister and Schreier.- 1. Frobenius' representation.- 2. Semidirect products.- 3. Subgroups of free groups are free.- 4. The calculus of presentations.- 5. The calculus of presentations (continued).- 6. The Reidemeister-Schreier method.- 7. Generalized free products.- IV Recursively presented groups, word problems and some applications of the Reidemeister-Schreier method.- 1. Recursively presented groups.- 2. Some word problems.- 3. Groups with free subgroups.- V Affine algebraic sets and the representative theory of finitely generated groups.- 1. Background.- 2. Some basic algebraic geometry.- 3. More basic algebraic geometry.- 4. Useful notions from topology.- 5. Morphisms.- 6. Dimension.- 7. Representations of the free group of rank two in SL(2,C).- 8. Affine algebraic sets of characters.- VI Generalized free products and HNN extensions.- 1. Applications.- 2. Back to basics.- 3. More applicatone.- 4. Some word, conjugacy and isomorphism problems.- VII Groups acting on trees.- 1. Basic definitions.- 2. Covering space theory.- 3. Graphs of groups.- 4. Trees.- 5. The fundamental group of a graph of groups.- 6. The fundamental group of a graph of groups (continued).- 7. Group actions and graphs of groups.- 8. Universal covers.- 9. The proof of Theorem 2.- 10. Some consequences of Theorem 2 and 3.- 11. The tree of SL2.
Series: Lectures in Mathematics Eth Zurich
Number Of Pages: 170
Published: 1st September 1993
Publisher: Birkhauser Verlag AG
Country of Publication: CH
Dimensions (cm): 23.39 x 15.6
Weight (kg): 0.26