Words Notes on Verbal Width in Groups

After a forty-year lull, the study of word-values in groups has sprung back into life with some spectacular new results in finite group theory. These are largely motivated by applications to profinite groups, including the solution of an old problem of Serre. This book presents a comprehensive account of the known results, both old and new. The more elementary methods are developed from scratch, leading to self-contained proofs and improvements of some classic results about infinite soluble groups. This is followed by a detailed introduction to more advanced topics in finite group theory, and a full account of the applications to profinite groups. The author presents proofs of some very recent results and discusses open questions for further research. This self-contained account is accessible to research students, but will interest all research workers in group theory.

Preface; 1. Generalities; 2. Verbally elliptic classes; 3. Words of infinite width; 4. Words and profinite groups; 5. Algebraic and analytic groups; Appendix; Bibliography; Index.

Spectacular new results in finite group theory can be accessed via the study of word-values in groups. This book explores fundamental questions about the behaviour of word-values in groups and demonstrates the power of this approach. An accessible treatment suitable for research students and experienced workers in group theory.

Dan Segal is Senior Research Fellow at All Souls College, Oxford, and Professor of Mathematics at the University of Oxford.