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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.

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