Topics in geometric group theory.

*(English)*Zbl 0965.20025
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.

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 |