eBook ISBN:  9781470402099 
Product Code:  MEMO/130/620.E 
List Price:  $47.00 
MAA Member Price:  $42.30 
AMS Member Price:  $28.20 
eBook ISBN:  9781470402099 
Product Code:  MEMO/130/620.E 
List Price:  $47.00 
MAA Member Price:  $42.30 
AMS Member Price:  $28.20 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 130; 1997; 117 ppMSC: Primary 20; 57
Diagram groups are groups consisting of spherical diagrams (pictures) over monoid presentations. They can be also defined as fundamental groups of the Squier complexes associated with monoid presentations. The authors show that the class of diagram groups contains some wellknown groups, such as the R. Thompson group \(F\). This class is closed under free products, finite direct products, and some other grouptheoretical operations. The authors develop combinatorics on diagrams similar to the combinatorics on words. This helps in finding some structure and algorithmic properties of diagram groups. Some of these properties are new even for R. Thompson's group \(F\). In particular, the authors describe the centralizers of elements in \(F\), prove that it has solvable conjugacy problem, and more.
ReadershipGraduate students and research mathematicians interested in group theory.

Table of Contents

Chapters

1. Introduction

2. Rewrite systems

3. Semigroup diagrams

4. Monoid pictures

5. Diagram groups

6. Squier’s complexes

7. Monoid presentations and the diagram groups

8. Diagram groups and group theoretic constructions

9. Diagram groups over complete presentations

10. Finitely presented diagram groups

11. Commutator subgroups of diagram groups

12. Asphericity

13. Recursive presentations of diagram groups

14. Computational complexity of the word problem in diagram groups

15. Combinatorics on diagrams

16. Different types of diagrams and finitely presented simple groups

17. Open problems


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Diagram groups are groups consisting of spherical diagrams (pictures) over monoid presentations. They can be also defined as fundamental groups of the Squier complexes associated with monoid presentations. The authors show that the class of diagram groups contains some wellknown groups, such as the R. Thompson group \(F\). This class is closed under free products, finite direct products, and some other grouptheoretical operations. The authors develop combinatorics on diagrams similar to the combinatorics on words. This helps in finding some structure and algorithmic properties of diagram groups. Some of these properties are new even for R. Thompson's group \(F\). In particular, the authors describe the centralizers of elements in \(F\), prove that it has solvable conjugacy problem, and more.
Graduate students and research mathematicians interested in group theory.

Chapters

1. Introduction

2. Rewrite systems

3. Semigroup diagrams

4. Monoid pictures

5. Diagram groups

6. Squier’s complexes

7. Monoid presentations and the diagram groups

8. Diagram groups and group theoretic constructions

9. Diagram groups over complete presentations

10. Finitely presented diagram groups

11. Commutator subgroups of diagram groups

12. Asphericity

13. Recursive presentations of diagram groups

14. Computational complexity of the word problem in diagram groups

15. Combinatorics on diagrams

16. Different types of diagrams and finitely presented simple groups

17. Open problems