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Combinatorial group theory. Reprint of the 1977 ed. (English) Zbl 0997.20037
Classics in Mathematics. Berlin: Springer. xiv, 339 p. (2001).
This is a reprint of the 1977 edition (Zbl 0368.20023) of this famous and very popular book, which became a desk copy for everybody who is dealing with combinatorial group theory. The authors describe with details two broad methods. The first part of the book (free groups and their subgroups, and generators and relations) is dedicated to the fundamental algorithms of Nielsen, Reidemeister, Schreier, Tietze and Whitehead. The authors prove the Freiheitssatz in different ways in order to show the power of fundamental methods. The second part is about geometric methods, free products and HNN extensions, and small cancellation theory. The complete bibliography (more than 1000 titles) well reflects the situation in the combinatorial group theory at the time when the book was published. Definitely, since the face of the combinatorial group theory has significantly changed (for instance, by works of A. Yu. Ol’shanskij and his students), this well-written book still is very functional and efficient.

20F05 Generators, relations, and presentations of groups
20-02 Research exposition (monographs, survey articles) pertaining to group theory
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20F06 Cancellation theory of groups; application of van Kampen diagrams
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)