Trees of cylinders and canonical splittings

2008/11/14 by Vincent Guirardel, Gilbert Levitt · 48 citations
Mathematics · #Abelian group #Automorphism #Combinatorics #Commensurability (mathematics) #Commutative property #Equivalence relation #Finite Group Theory Research #Finitely-generated abelian group #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Invariant (physics) #Isogeny #Mathematical physics #Mathematics #Pure mathematics #Transitive relation #math.GR #math.GT #msc:20E06 #msc:20E08 #msc:20F65 #msc:20F67

paper · pdf · doi:10.2140/gt.2011.15.977

published in Geometry & Topology 15(2), 977-1012 (Mathematical Sciences Publishers) · 38 pages, 2 figures. Reference update

arxiv created 2008/11/14 · openalex publication_date 2011/06/22 · arxiv updated 2016/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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.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).

Citations

• JSJ-splittings for finitely presented groups over slender groups
• Cut points and canonical splittings of hyperbolic groups
• JSJ-Decompositions of finitely presented groups and complexes of groups
• Acylindrical accessibility for groups
• On uniqueness of JSJ decompositions of finitely generated groups
• RELATIVELY HYPERBOLIC GROUPS
• Cyclic Splittings of Finitely Presented Groups and the Canonical JSJ Decomposition

Cited by

• A gluing theorem for negatively curved complexes
• Recognizing a relatively hyperbolic group by its Dehn fillings
• Property R_∞ for groups with infinitely many ends