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Core and intersection number for group actions on trees. (Cœur et nombre d’intersection pour les actions de groupes sur les arbres.) (French. English summary) Zbl 1110.20019
Given two splittings of a group $$G$$ as graphs of groups, the Bass-Serre theory produces two actions of $$G$$ on two simplicial trees. In this paper, the author introduces a geometric construction of a ‘convex core’ for the diagonal action of $$G$$ on the product of these trees. More generally, he associates a convex core to the product of two actions of a group on $$\mathbb{R}$$-trees. This construction generalizes and unifies the following notions: the intersection number of simple closed curves and of measured foliations on surfaces; the Culler-Levitt-Shalen convex core associated to two actions on $$\mathbb{R}$$-trees dual to measured foliations; the Fujiwara-Papasoglu JSJ decomposition providing a surface in the product of two simplicial trees, which is also related to an intersection number defined by Scott and Swarup.
As an application of the notions introduced, the author proves that an irreducible automorphism of a free group whose stable and unstable trees are geometric is induced by a pseudo-Anosov homeomorphism of the surface. The author notes this last result has also been obtained by Mosher and Handel, using other methods.
An English version of the paper under review is available on arxiv,
http://arxiv.org/abs/math.GR/0407206.

##### MSC:
 20E08 Groups acting on trees 20F65 Geometric group theory 20E05 Free nonabelian groups 20E36 Automorphisms of infinite groups
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##### References:
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