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Trees of cylinders and canonical splittings

Vincent Guirardel and Gilbert Levitt

Geometry & Topology 15 (2011) 977–1012

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 Tc. 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, Tc has very strong compatibility properties (two trees are compatible if they have a common refinement).

JSJ decomposition, canonical decomposition, amalgamated free product
Mathematical Subject Classification 2000
Primary: 20E08
Secondary: 20F65, 20F67, 20E06
Received: 10 December 2008
Accepted: 29 March 2011
Published: 22 June 2011
Proposed: Martin Bridson
Seconded: Benson Farb, Danny Calegari
Vincent Guirardel
Institut de Mathématiques de Toulouse
Université de Toulouse CNRS (UMR 5219)
118 route de Narbonne
F-31062 Toulouse cedex 9
Institut de Recherche Mathématiques de Rennes
Université de Rennes 1 CNRS (UMR 6625)
263 avenue du General Leclerc
CS 74205
35042 Rennes Cedex
Gilbert Levitt
Laboratoire de Mathématiques Nicolas Oresme
Université de Caen CNRS (UMR 6139)
BP 5186
F-14032 Caen Cedex