Trees of cylinders and canonical splittings

Vincent Guirardel (1) , Gilbert Levitt (2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 LMNO - Laboratoire de Mathématiques Nicolas Oresme

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).

Domaines: Théorie des groupes [math.GR]; Topologie géométrique [math.GT]

Vincent Guirardel, Gilbert Levitt. Trees of cylinders and canonical splittings. Geometry and Topology, 2011, 15, pp.977-1012. ⟨10.2140/gt.2011.15.977⟩. ⟨hal-00353138⟩