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Diagram Groups
Victor Guba, Vologda State Pedagogical Institute, Russia, and Mark Sapir, University of Nebraska, Lincoln, NE
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Memoirs of the American Mathematical Society
1997; 117 pp; softcover
Volume: 130
ISBN-10: 0-8218-0639-4
ISBN-13: 978-0-8218-0639-5
List Price: US$47 Individual Members: US$28.20
Institutional Members: US\$37.60
Order Code: MEMO/130/620

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 well-known groups, such as the R. Thompson group $$F$$. This class is closed under free products, finite direct products, and some other group-theoretical 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.

• Introduction
• Rewrite systems
• Semigroup diagrams
• Monoid pictures
• Diagram groups
• Squier's complexes
• Monoid presentations and the diagram groups
• Diagram groups and group theoretic constructions
• Diagram groups over complete presentations
• Finitely presented diagram groups
• Commutator subgroups of diagram groups
• Asphericity
• Recursive presentations of diagram groups
• Computational complexity of the word problem in diagram groups
• Combinatorics on diagrams
• Different types of diagrams and finitely presented simple groups
• Open problems