Topics in geometric group theory. (English) Zbl 0965.20025

de la Harpe, Pierre

Chicago Lectures in Mathematics. Chicago: The University of Chicago Press. vi, 310 p. (2000).

Pierre de la Harpe’s “Topics in Geometric Group Theory” is attractive for a number of reasons: (1) It is full of explicit and interesting examples of finitely-generated infinite groups which illustrated important features of the landscape of geometric group theory; (2) it has an extensive list of references (30 pages); (3) it has numerous exercises, problems, open problems, and complements for the reader; (4) it has four pages of research problems.
Highlights (chapter by chapter) are Pólya’s recurrence theorem for random walks; Euclidean spaces; Klein’s criterion for free groups (the Table-Tennis Lemma); B. H. Neumann’s construction of uncountably many distinct two-generator groups; Cayley graphs and quasi-isometries; Poincaré’s theorem on fundamental polygons (polyhedra); fundamental groups and Riemannian curvature; growth of finitely-generated groups; and Grigorchuk’s first group of intermediate growth. The author’s coverage of geometric group theory is idiosyncratic, but the author gives a nice list of explicit references for omitted topics.

Reviewer: J.W.Cannon (Provo)

MSC:

20F65	Geometric group theory
20-02	Research exposition (monographs, survey articles) pertaining to group theory
20F05	Generators, relations, and presentations of groups
20E07	Subgroup theorems; subgroup growth

Keywords:

examples; finitely generated infinite groups; geometric group theory; bibliography; open problems; free groups; two-generator groups; Cayley graphs; quasi-isometries; fundamental groups; growth of finitely-generated groups; groups of intermediate growth