## Geometry & Topology

### Trees of cylinders and canonical splittings

#### Abstract

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, coelementarity in a relatively hyperbolic group, or commutation in a commutative transitive group), we define a tree of cylinders $Tc$. This tree only depends on the deformation space of $T$; in particular, it is invariant under automorphisms of $G$ if $T$ is a JSJ splitting. We thus obtain $Out(G)$–invariant cyclic or abelian JSJ splittings. Furthermore, $Tc$ has very strong compatibility properties (two trees are compatible if they have a common refinement).

#### Article information

Source
Geom. Topol., Volume 15, Number 2 (2011), 977-1012.

Dates
Accepted: 29 March 2011
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513732309

Digital Object Identifier
doi:10.2140/gt.2011.15.977

Mathematical Reviews number (MathSciNet)
MR2821568

Zentralblatt MATH identifier
1272.20026

#### Citation

Guirardel, Vincent; Levitt, Gilbert. Trees of cylinders and canonical splittings. Geom. Topol. 15 (2011), no. 2, 977--1012. doi:10.2140/gt.2011.15.977. https://projecteuclid.org/euclid.gt/1513732309

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