The geometry of abstract groups and their splittings. (Q1431090)

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).