The geometry of abstract groups and their splittings.

Wall, Charles Terence Clegg. "The geometry of abstract groups and their splittings.." Revista Matemática Complutense 16.1 (2003): 5-101. <http://eudml.org/doc/44354>.

@article{Wall2003,
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 L-tree, and the L-tree of a valued field, 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.},
author = {Wall, Charles Terence Clegg},
journal = {Revista Matemática Complutense},
keywords = {Teoría de grupos; Acción de grupo; Arboles topológicos; Grupos abstractos; splittings of groups; groups acting on trees; Poincaré duality groups; hyperbolic groups; convergence groups; free products with amalgamation; ends of groups; graphs of groups; JSJ decompositions},
language = {eng},
number = {1},
pages = {5-101},
title = {The geometry of abstract groups and their splittings.},
url = {http://eudml.org/doc/44354},
volume = {16},
year = {2003},
}

TY - JOUR
AU - Wall, Charles Terence Clegg
TI - The geometry of abstract groups and their splittings.
JO - Revista Matemática Complutense
PY - 2003
VL - 16
IS - 1
SP - 5
EP - 101
AB - 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 L-tree, and the L-tree of a valued field, 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.
LA - eng
KW - Teoría de grupos; Acción de grupo; Arboles topológicos; Grupos abstractos; splittings of groups; groups acting on trees; Poincaré duality groups; hyperbolic groups; convergence groups; free products with amalgamation; ends of groups; graphs of groups; JSJ decompositions
UR - http://eudml.org/doc/44354
ER -