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The geometry of abstract groups and their splittings. (English) Zbl 1059.20039

This is a survey on the part of geometric and combinatorial group theory centering around splittings of abstract groups and their applications, with a strong emphasis on a geometric and topological point of view; it passes from historical origins and classical theorems (Kurosh 1937, Grushko 1940, the definition of Hanna Neumann of free products with amalgamation from 1948, HNN-extensions form 1949) over Stallings’ fundamental results on groups with more than one end and the Bass-Serre theory of groups acting on trees to significant recent developments as Gromov’s notion of word hyperbolic group, the breakthrough of Rips on groups acting on \(\mathbb{R}\)-trees and the solution of the problem of convergence groups on the circle. The paper contains a wealth of information on the various developments and the relevant literature (more than 10 pages of references).
The contents of the chapters are as follows: 1. Preliminaries; 2. Actions of groups on trees (graphs of groups and groups acting on trees, structure of \(G\)-trees, length functions); 3. Stallings’ theorem (ends of groups, virtually free groups, splittings over finite groups); 4. Bounds on splittings (accessibility, uniqueness of splittings); 5. Poincaré duality groups in dimension 2; 6. Splittings over infinite groups (JSJ theorems); 7. Geometry of groups (quasi-isometry and hyperbolic groups, the boundary of a hyperbolic group, convergence groups); 8. \(\mathbb{R}\)-trees (Rips’ theorem and applications); 9. Further splitting theorems (splittings over two-ended subgroups, actions on CAT(0) cube complexes, coarse geometry); 10. \(PD^2\) and \(PD^3\) complexes and groups (a survey to which extent it is possible to push through the Thurston program for Poincaré duality complexes and pairs in dimension three).

MSC:

20F65 Geometric group theory
20-02 Research exposition (monographs, survey articles) pertaining to group theory
20E08 Groups acting on trees
20F67 Hyperbolic groups and nonpositively curved groups
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20J05 Homological methods in group theory
57M07 Topological methods in group theory
57P10 Poincaré duality spaces
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