Topics in Combinatorial Group Theory

ISBN-13: 9783764329211 / Angielski / Miękka / 1993 / 170 str.

Gilbert Baumslag; G. Baumslag

Opis

Combinatorial group theory is a loosely defined subject, with close connections to topology and logic. The objective of combinatorial group theory is the systematic development of algebraic techniques to settle such questions. This title features notes that deals with many topics in combinatorial group theory.

Spis treści

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.