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Trees of cylinders and canonical splittings. (English) Zbl 1272.20026

Summary: 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 \(\mathrm{Out}(G)\)-invariant cyclic or Abelian JSJ splittings. Furthermore, \(T_c\) has very strong compatibility properties (two trees are compatible if they have a common refinement).

MSC:

20E08 Groups acting on trees
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20F65 Geometric group theory
20F67 Hyperbolic groups and nonpositively curved groups
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