The geometry of abstract groups and their splittings

Abstract

A survey of splitting theorems for abstract groups and their applications. Topics covered include preliminaries, early results, Bass-Serre theory, the structure of G-trees, Serre's applications to SL2 and length functions. Stallings' theorem, results about accessibility and bounds for splittability. Duality groups and pairs; results of Eckmann and collaborators on PD2 groups. Relative ends, the JSJ theorems and the splitting results of Kropholler and Roller on PDn groups. Notions of quasi isometry, of hyperbolic group, and of its boundary. We recall that convergence groups on the circle are Fuchsian, and survey results relating properties of the action of a hyperbolic group on its boundary to the structure of the group. Types of isometric action of a group on a _-tree, and the _ tree of a valued _eld, with mention of the applications made by Culler, Shalen and Morgan. Rips' theorem, and some of its applications. Splittings over 2-ended groups and work of Sela and Bowditch, more general splitting theorems, characterisations of groups by their coarse geometry. Finally we survey the extent to which it is possible to push through the Thurston programme for PD3 complexes and pairs: despite many advances, there remain more conjectures than theorems.