de la Harpe, Pierre 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. Reviewer: J.W.Cannon (Provo) Cited in 8 ReviewsCited in 405 Documents 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 × Cite Format Result Cite Review PDF Online Encyclopedia of Integer Sequences: a(n) = 3*(2*n)!/((n+2)!*(n-1)!). Fourth convolution of Catalan numbers: a(n) = 4*binomial(2*n+3,n)/(n+4). Number of connected permutations of [1..n] (those not fixing [1..j] for 0 < j < n). Also called indecomposable permutations, or irreducible permutations. Expansion of (1 + 2*x + x^2)/(1 - 10*x + x^2). Expansion of (1+x)^2/(1-18*x+x^2). Expansion of (1+2*x+x^2)/(1-26*x+x^2). Expansion of (1+2*x+x^2)/(1-34*x+x^2). Expansion of (1+2*x+x^2)/(1-42*x+x^2). Expansion of (1+2*x+x^2)/(1-50*x+x^2). Expansion of (1+2*x+x^2)/(1-58*x+x^2). Expansion of (1+2*x+x^2)/(1-66*x+x^2). Expansion of (1+2*x+x^2)/(1-74*x+x^2). a(n) = 6*a(n-1) - a(n-2). Triangle of Mahonian numbers T(n,k): coefficients in expansion of Product_{i=0..n-1} (1 + x + ... + x^i), where k ranges from 0 to A000217(n-1). Also enumerates permutations by their major index. Coordination sequence for planar net 3.6.3.6. Spherical growth function for a certain reflection group in plane. Number of subgroups of index n in free group of rank 3. Array T(n,k) = number of subgroups of index k in free group of rank n, read by antidiagonals. Number of subgroups of index n in free group of rank 4. Number of subgroups of index 3 in free group of rank n+1. Number of subgroups of index 4 in free group of rank n+1. Layer counting sequence for hyperbolic tessellation by cuspidal triangles of angles (Pi/3,Pi/3,0) (this is the classical modular tessellation). Number of subgroups of index n in free group of rank n. The circle problem: number of points (x,y) in square lattice with x^2 + y^2 <= n. G.f.: (1+3*x+2*x^2)/((1-x)*(1-2*x^2)). a(0)=1, a(n) = 2*Fibonacci(n+4) - 6. Spherical growth series for modular group. Spherical growth series for Z as generated by {2, 3}. Spherical growth series for pair of groups, one Gromov hyperbolic, the other not. Growth series for Heisenberg group. Erroneous version of A008579. Growth series for fundamental group of orientable closed surface of genus 2. Growth series for fundamental group of orientable closed surface of genus 3. Growth series for fundamental group of orientable closed surface of genus 4. Growth series for fundamental group of orientable closed surface of genus 5. Growth series for fundamental group of orientable closed surface of genus 6. Growth series for fundamental group of orientable closed surface of genus 7. Growth series for fundamental group of orientable closed surface of genus 8. Growth series for fundamental group of orientable closed surface of genus 9. Growth series for fundamental group of orientable closed surface of genus 10. Growth series for fundamental group of orientable closed surface of genus 11. Growth series for fundamental group of orientable closed surface of genus 12. Expansion of g.f. (1 - 2*x^2 - 3*x^3)/((1 - x^3)*(1 - 2*x)).