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Infinite groups as geometric objects (after Gromov). (English) Zbl 0764.57003
Ergodic theory, symbolic dynamics, and hyperbolic spaces, Lect. Workshop Hyperbolic Geom. Ergodic Theory, Trieste/Italy 1989, 299-314 (1991).
[For the entire collection see Zbl 0743.00040.]
Following M. Gromov [Proc. Int. Congr. Math., Warszawa 1983, Vol. 1, 385-392 (1984; Zbl 0599.20041)], the main point of the authors is to show that a geometrization of algebraic problems may be fruitful: given an algebraic problem, translate it into geometry and make it “visible”. The algebraic problem of their concern is to understand finitely generated groups. Crucial tool for that is the using of Cayley graphs and the notion of quasi-isometry. In particular, they bring a list of “geometric properties” of groups (invariant under quasi-isometry): degree of polynomial growth, being of finite presentation, being virtually nilpotent (or Abelian, or cyclic), amenability, being hyperbolic [see M. Gromov, Essays in group theory, Publ., Math. Sci. Res. Inst. 8, 75-263 (1987; Zbl 0634.20015)].

57M07 Topological methods in group theory
57M05 Fundamental group, presentations, free differential calculus
57S30 Discontinuous groups of transformations
53C99 Global differential geometry
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)