# JSJ-Decompositions of finitely presented groups and complexes of groups

@article{Fujiwara2005JSJDecompositionsOF, title={JSJ-Decompositions of finitely presented groups and complexes of groups}, author={Koji Fujiwara and Panos Papasoglu}, journal={Geometric \& Functional Analysis GAFA}, year={2005}, volume={16}, pages={70-125} }

Abstract.A JSJ-splitting of a group G over a certain class of subgroups is a graph of groups decomposition of G which describes all possible decompositions of G as an amalgamated product or an HNN extension over subgroups lying in the given class. Such decompositions originated in 3-manifold topology. In this paper we generalize the JSJ-splitting constructions of Sela, Rips–Sela and Dunwoody–Sageev, and we construct a JSJ-splitting for any finitely presented group with respect to the class of…

## 73 Citations

JSJ-Decompositions of Coxeter Groups over Virtually

- Mathematics
- 2008

The idea of “JSJ-decompositions” for 3-manifolds began with work of Waldhausen and was developed later through work of Jaco, Shalen and Johansen. It was shown that there is a finite collection of…

JSJ-Decompositions of Coxeter Groups over Virtually Abelian Splittings

- Mathematics
- 2008

The idea of “JSJ-decompositions” for 3-manifolds began with work of Waldhausen and was developed later through work of Jaco, Shalen and Johansen. It was shown that there is a finite collection of…

JSJ Decompositions of Coxeter Groups

- Mathematics
- 2008

The idea of "JSJ-decompositions" for 3-manifolds began with work of Waldhausen and was developed later through work of Jaco, Shalen and Johansen. It was shown that there is a finite collection of…

Strong accessibility for finitely presented groups

- Mathematics
- 2013

A hierarchy of a group is a rooted tree of groups obtained by iteratively passing to vertex groups of graphs of groups decompositions. We define a (relative) slender JSJ hierarchy for (almost)…

Jsj Decompositions of Groups

- Mathematics
- 2018

This is an account of the theory of JSJ decompositions of finitely generated groups, as developed in the last twenty years or so. We give a simple general definition of JSJ decompositions (or rather…

Splittings of non-finitely generated groups

- Mathematics
- 2012

In geometric group theory one uses group actions on spaces to gain information about groups. One natural space to use is the Cayley graph of a group. The Cayley graph arguments that one encounters…

On uniqueness of JSJ decompositions of finitely generated groups

- Mathematics
- 2003

AbstractWe give an example of two JSJ decompositions of a group that are not
related by conjugation, conjugation of edge-inclusions, and slide
moves. This answers the question of Rips and Sela stated…

Quasi-isometries between groups with two-ended splittings

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 2016

Abstract We construct a ‘structure invariant’ of a one-ended, finitely presented group that describes the way in which the factors of its JSJ decomposition over two-ended subgroups fit together. For…

Trees of cylinders and canonical splittings

- Mathematics
- 2011

Let T be a tree with an action of a finitely generated group G. Given a suitable equivalence relation on the set of edge stabilizers of T (such as commensurability, co-elementarity in a relatively…

Computing JSJ decompositions of hyperbolic groups

- Mathematics, Computer Science
- 2016

An algorithm that computes Bowditch's canonical JSJ decomposition of a given one‐ended hyperbolic group over its virtually cyclic subgroups with infinite centre is presented.

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