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**Geometric group theory. Volume 2: Asymptotic invariants of infinite groups. Proceedings of the symposium held at the Sussex University, Brighton, July 14-19, 1991.**
*(English)*
Zbl 0841.20039

London Mathematical Society Lecture Note Series. 182. Cambridge: Cambridge University Press. vii, 295 p. (1993).

This ground breaking paper (a book) may be enthusiastically recommended to researchers in several subjects – combinatorial group theory, geometry and topology of manifolds of non-positive curvature, dynamics, etc. It provides a fascinating look at finitely generated groups, which continues the author’s study of infinite groups from a geometric viewpoint started in his previous works [Ann. Math. Stud. 97, 183-213 (1981; Zbl 0467.53035); Proc. ICM, Warsawa 1983, Vol. I, 385-392 (1984; Zbl 0599.20041), MSRI Publ., Math. Sci. Res. Inst. 8, 75-263 (1987; Zbl 0634.20015), see also D. Epstein, J. Cannon, D. Holt, S. Levy, M. Paterson and W. Thurston, Word processing in groups (1992; Zbl 0764.20017)]. The author has again opened up new vistas that geometers and geometric group theorists may explore in the years to come.

The major section titles are: (1) Asymptotic methods (thickening, asymptotic connectedness, contractibility, large scale dimension, (co)homology and width); (2) Asymptotic cones (in particular, for Lie groups and lattices; geometry of such cones and the space of the word metrics); (3) Distorsion (with intrinsic geometry of subgroups and submanifolds; polynomial, exponential and nonrecursive distortion); (4) Topology of balls (indiscrete and highly non-contractible in the conventional sense spaces); (5) Filling invariants (isoperimetric and isodiametric functions, area filling in nilpotent and solvable groups, filling length, radius and Morse landscape, filling on the large scale and volume distortion of subspaces); (6) Semi-hyperbolic spaces \((K \leq 0\), coning and combing, rank and geometry of flags, Tits boundary and Morse landscape at infinity, minimal surfaces and norms on homology, group actions with \(K \leq 0\)); (7) Hyperbolic groups (groups and curvature, hierarchy in the hyperbolic ranks, pinching and conformal geometry at infinity, round trees, Kähler and anti-Kähler groups, uniform embeddings and \(T\)-amenability); (8) \(L_p\)-Cohomology (with Appendix on conformal hyperbolic groups and spaces); (9) Finitely presented groups. This list of headings hardly indicates the breadth of discussion.

The major section titles are: (1) Asymptotic methods (thickening, asymptotic connectedness, contractibility, large scale dimension, (co)homology and width); (2) Asymptotic cones (in particular, for Lie groups and lattices; geometry of such cones and the space of the word metrics); (3) Distorsion (with intrinsic geometry of subgroups and submanifolds; polynomial, exponential and nonrecursive distortion); (4) Topology of balls (indiscrete and highly non-contractible in the conventional sense spaces); (5) Filling invariants (isoperimetric and isodiametric functions, area filling in nilpotent and solvable groups, filling length, radius and Morse landscape, filling on the large scale and volume distortion of subspaces); (6) Semi-hyperbolic spaces \((K \leq 0\), coning and combing, rank and geometry of flags, Tits boundary and Morse landscape at infinity, minimal surfaces and norms on homology, group actions with \(K \leq 0\)); (7) Hyperbolic groups (groups and curvature, hierarchy in the hyperbolic ranks, pinching and conformal geometry at infinity, round trees, Kähler and anti-Kähler groups, uniform embeddings and \(T\)-amenability); (8) \(L_p\)-Cohomology (with Appendix on conformal hyperbolic groups and spaces); (9) Finitely presented groups. This list of headings hardly indicates the breadth of discussion.

Reviewer: B.N.Apanasov (Norman)

### MSC:

20F65 | Geometric group theory |

20F05 | Generators, relations, and presentations of groups |

20-02 | Research exposition (monographs, survey articles) pertaining to group theory |

57M07 | Topological methods in group theory |