Combinatorial Group Theory

Open Problems on Infinite Groups

Ilya Kapovich, Alexei Myasnikov, Vladimir Shpilrain

• Publisher: De Gruyter
• Year: 2026
• Topic: Mathematics
• eBook ISBN:
  9783112208380
• Subtopic: Algebra
• Index: Mathematics

This is an annotated collection of over 200 open problems in combinatorial group theory. One can say that most of the problems are closer in spirit to the "purely combinatorial" group theory. However, the authors included several problems with geometric flavor where they felt this was natural. 
Approximately 20 years ago, a new direction in group theory began to emerge at the interface with theoretical computer science. In recognition of this, the given collection includes problems that are not only significant in combinatorial group theory but also have direct relevance to this emerging area, such as the Post correspondence problem for groups, which is gaining momentum in contemporary research. 
The "flagship" section includes problems about free groups; these have been at the center of combinatorial group theory for over a century, and they are getting ever more popular due to newly discovered connections with the theoretical computer science mentioned above.
Other classes of groups that the authors cover include hyperbolic groups, solvable groups, groups of matrices, and others. The target audience of the book is very broad: from PhD students to senior researchers.

Table of contents

• Title Page
• Copyright
• Contents
• Frontmatter
• Contents
• 1 Outstanding problems
• 2 Free groups
• 3 One-relator groups
• 4 Finitely presented groups
• 5 Mapping class groups
• 6 Growth
• 7 Equations in and over groups
• 8 Algorithmic problems
• 9 Complexity of algorithms
• 10 Groups of matrices
• 11 Hyperbolic and automatic groups
• 12 Nilpotent groups
• 13 Metabelian groups
• 14 Solvable groups
• 15 Overview of recent progress in classical areas
• 16 Hall of fame