# Diagram Groups

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*Victor Guba; Mark Sapir*

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.

#### Table of Contents

# Table of Contents

## Diagram Groups

- Contents vii8 free
- 1 Introduction 110 free
- 2 Rewrite Systems 716 free
- 3 Semigroup Diagrams 918
- 4 Monoid Pictures 2130
- 5 Diagram Groups 3140
- 6 Squier's Complexes 3342
- 7 Monoid Presentations and The Diagram Groups 3948
- 8 Diagram Groups and Group Theoretic Constructions 4554
- 9 Diagram Groups over Complete Presentations 4958
- 10 Finitely Presented Diagram Groups 5766
- 11 Commutator Subgroups of Diagram Groups 6271
- 12 Asphericity 6473
- 13 Recursive Presentations of Diagram Groups 6675
- 14 Computational Complexity of the Word Problem in Diagram Groups 7281
- 15 Combinatorics on Diagrams 7483
- 16 Different Types of Diagrams and Finitely Presented Simple Groups 97106
- 17 Open Problems 110119