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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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IV. 10. Geometric and Combinatorial Group Theory

Martin R. Bridson

1 What Are Combinatorial and Geometric Group Theory?

Groups and geometry are ubiquitous in mathematics, groups because the symmetries (or AUTOMORPHISMS [I.3 §4.1]) of any mathematical object in any context form a group and geometry because it allows one to think intuitively about abstract problems and to organize families of objects into spaces from which one may gain some global insight.

The purpose of this article is to introduce the reader to the study of infinite, discrete groups. I shall discuss both the combinatorial approach to the subject that held sway for much of the twentieth century and the more geometric perspective that has led to an enormous flowering of the subject ...

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