Geometry & Topology

Trees of cylinders and canonical splittings

Vincent Guirardel and Gilbert Levitt

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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).

Article information

Geom. Topol., Volume 15, Number 2 (2011), 977-1012.

Received: 10 December 2008
Accepted: 29 March 2011
First available in Project Euclid: 20 December 2017

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Zentralblatt MATH identifier

Primary: 20E08: Groups acting on trees [See also 20F65]
Secondary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 20F67: Hyperbolic groups and nonpositively curved groups 20E06: Free products, free products with amalgamation, Higman-Neumann- Neumann extensions, and generalizations

JSJ decomposition canonical decomposition amalgamated free product


Guirardel, Vincent; Levitt, Gilbert. Trees of cylinders and canonical splittings. Geom. Topol. 15 (2011), no. 2, 977--1012. doi:10.2140/gt.2011.15.977.

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