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, co-elementarity 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(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).

38 pages, 2 figures. Reference update

References in corpus (2)

• Whitehead moves for G-trees
• Scott and Swarup's regular neighbourhood as a tree of cylinders

Cited by in corpus (24)

• Homogeneity in the free group
• JSJ decompositions: definitions, existence, uniqueness. I: The JSJ deformation space
• Quasi-isometries Between Groups with Two-Ended Splittings
• JSJ decompositions: definitions, existence, uniqueness. II. Compatibility and acylindricity
• Recognizing a relatively hyperbolic group by its Dehn fillings
• McCool groups of toral relatively hyperbolic groups
• Deciding Isomorphy using Dehn fillings, the splitting case
• Quasi-isometry classification of RAAGs that split over cyclic subgroups
• Free-by-cyclic groups, automorphisms and actions on nearly canonical trees
• Detecting geometric splittings in finitely presented groups
• Conformal dimension of hyperbolic groups that split over elementary subgroups
• On Tarski's problem for virtually free groups
• Algebraic & definable closure in free groups
• Quasi-Isometries for certain Right-Angled Coxeter Groups
• Scott and Swarup's regular neighbourhood as a tree of cylinders
• Measure equivalence rigidity of $\mathrm{Out}(F_N)$
• Elementary equivalence vs commensurability for hyperbolic groups
• A gluing theorem for negatively curved complexes
• Vertex finiteness for splittings of relatively hyperbolic groups
• Quasi-isometry Invariants from Decorated Trees of Cylinders of Two-Ended JSJ Decompositions
• Forking and JSJ decompositions in the free group II
• Cannon--Thurston maps for CAT(0) groups with isolated flats
• Towers and the first-order theory of hyperbolic groups
• Cocycle superrigidity from higher rank lattices to $\mathrm{Out}(F_N)$