#### Volume 15, issue 2 (2011)

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Trees of cylinders and canonical splittings

### Vincent Guirardel and Gilbert Levitt

Geometry & Topology 15 (2011) 977–1012
##### 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 ${T}_{c}$. 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 $Out\left(G\right)$–invariant cyclic or abelian JSJ splittings. Furthermore, ${T}_{c}$ has very strong compatibility properties (two trees are compatible if they have a common refinement).

##### Keywords
JSJ decomposition, canonical decomposition, amalgamated free product
##### Mathematical Subject Classification 2000
Primary: 20E08
Secondary: 20F65, 20F67, 20E06