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Groups St Andrews 2017 in Birmingham

C. M. Campbell, C. W. Parker, M. R. Quick, E. F. Robertson, and C. M. Roney-Dougal, editors
Cambridge University Press
Publication Date: 
Number of Pages: 
London Mathematical Society Lecture Note Series
[Reviewed by
Charles Traina
, on
Every four years leading researchers in group theory meet to survey recent developments in all aspects of their field.  The present volume is the tenth in this series and consists of twenty-one papers on the following topics: generation of finite simple groups, block theory, fusion systems, algebraic groups, one-relator groups, geometric group theory, and Beauville groups. 
We give a brief description of one of the articles, "One-Relator Groups: An Overview" by Gilbert Baumslag, Benjamin Fine, and Gerhard Rosenberger.  The article was selected by the reviewer because it is based on work in one relator group theory of Dr. Wilhelm Magnus, the reviewer's thesis advisor. It also indicates the importance of the work of Max Dehn, Magnus' thesis advisor.
In the 1985 meeting of Groups St. Andrews, Gilbert Baumslag gave a course on one-relator groups which provided a view of the subject up to that point in time. This paper looks at the large amount of work done over the past three decades. Connections with surface groups and elementary theory, and a description of the surface group conjecture and the Gromov conjecture on surface groups are discussed. The authors discuss the solution by Wise of Baumslag's residual finiteness conjecture and also a new Baumslag conjecture on virtually free-by-cyclic groups. Various amalgam decompositions of one-relator groups and the Baumslag-Shalen conjectures are additionally examined. A number of open problems in one-relator group theory and their statuses are discussed. The final discussion is an introduction to a concept called plainarity based on the Magnus breakdown of a one-relator group. This may provide a systematic approach to the problems in the setting of one-relator groups and indicates the importance of the fundamental work done by Wilhelm Magnus in one-relator group theory. 
This volume does what the editors intended: it provides a ``snapshot of the state of the art in group theory." It can serve as a reference work for group theorists, and since many of the articles include open problems, can serve as a source of research topics as well. 
Charles Traina is a professor of mathematics at St. John's University, Jamaica N.Y. His research areas are group theory and measure theory.