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 $T_c$. 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).