Memoirs of the American Mathematical Society 1997; 117 pp; softcover Volume: 130 ISBN10: 0821806394 ISBN13: 9780821806395 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 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. Readership Graduate students and research mathematicians interested in group theory. Table of Contents  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
