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Combinatorial group theory: a topological approach. (English) Zbl 0697.20001

London Mathematical Society Student Texts, 14. Cambridge etc.: Cambridge University Press. x, 310 p. £30.00/hpb; £11.95/pbk; $ 49.50/hbk; $ 19.95/pbk (1989).
This book is a substantially expanded and up-dated version of the notes bearing the same title, published in 1978 in the series Queen Mary College Mathematics Notes (Zbl 0389.20024). It manages to contain a large proportion of the core results of the theory of group presentations in terms of generators and relations (for example Grushko’s theorem and relatively recent results concerning the fixed subgroup of an automorphism of a finitely generated free group), while maintaining as a coherent central theme the hierarchy of “subgroup theorems” (of Schreier, Kurosh, Karrass-Solitar), culminating in the Bass-Serre theory of groups acting on trees. The development proceeds carefully, and fairly leisurely, starting essentially from zero, with strong emphasis on topological methods. However various kinds of proof - algebraic and graph-theoretical as well as topological - are given, or indicated, for some of the main theorems, with resemblances, if any, noted. The topology is explicated in a painstaking manner, and several interesting side applications are given along the way - for instance the Fundamental Theorem of Algebra and the Borsuk-Ulam “Ham-Sandwich Theorem”. Indeed the book would serve well as the text for a course in topology alone.
The contents are briefly as follows: Chapter 1 introduces the basic group-theoretical constructs: free groups, presentations, free products, pushouts and amalgamated free products, and HNN extensions. After setting the topological scene in Chapters 2 and 3 with the appropriate point-set topology, the definitions of paths and homotopies, and of groupoids and direct limits, the author introduces in Chapter 4 the fundamental groupoid and fundamental group of a space, proves Van Kampen’s theorem using groupoids, and establishes the essentials of covering-space theory, looking incidentally at applications (see above) as well as certain topological “pathologies”. In Chapters 5 and 6, which parallel to some extent Chapter 4, the theory of covering spaces is further elaborated, in the combinatorial context of complexes (i.e. of 2-dimensional CW- complexes denuded of their topology) as well as that of topological spaces. In chapter 7 this theory is exploited to yield the Schreier and Kurosh subgroup theorems, and, in Chapter 8, a purely combinatorial treatment of the Bass-Serre theory of the structure of groups acting on trees, which is then followed by several applications.
Thus the first eight chapters have the unifying (though not exhaustive) theme of covering-space theory and subgroup theorems. Chapter 9 represents a departure from this theme: here the author sketches in very readable form, again essentially from the beginning, algorithmic decision theory as it pertains to groups, using notably the approach via “modular machines” due to Aanderaa and himself. The final Chapter 10 provides a quick overview of “small-cancellation theory”, and other topics.
Of the other comparable texts on combinatorial group theory, the encyclopedic monographs by Magnus-Karrass-Solitar, and by Lyndon-Schupp are overall more algebraic in their methodology (although the latter employs “complexes” and contains sketches of topological proofs of some results), while that of Johnson overlaps but little. The book of Stillwell, though more radically topological in its approach, is not so much concerned with precision as with giving the flavour (and history) of the whole field in its topological aspects. Massey’s book on algebraic topology is rather more exclusively concerned with topology, and, the recent book of Dicks and Dunwoody, although it treats of Bass-Serre theory, has a quite different flavour and emphasis. Finally, the book of Serre is much less concerned with group theory for its own sake than is Cohen’s.
The present book can be strongly recommended to beginning graduate students looking for a largely self-contained, readable and coherent account of the basic preoccupations of combinatorial group theory, and also to group-theoretical experts in need of a topological corrective to their algebraic bias, or a detailed introduction to one of the topological sources of their subject, or merely a good modern reference for many of the central results of the theory. It is highly gratifying to have the close connections between topology and the theory of group presentations expounded as lucidly and accessibly as in the present text.
Reviewer: R.G.Burns

MSC:

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
20F05 Generators, relations, and presentations of groups
57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
20E07 Subgroup theorems; subgroup growth
55-02 Research exposition (monographs, survey articles) pertaining to algebraic topology
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
57M05 Fundamental group, presentations, free differential calculus
57M10 Covering spaces and low-dimensional topology

Citations:

Zbl 0389.20024
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