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Rips induction: index of the dual lamination of an $$\mathbb R$$-tree. (English) Zbl 1336.20033
Summary: Let $$T$$ be an $$\mathbb R$$-tree in the boundary of the outer space $$\mathrm{CN}_N$$, with dense orbits. The $$\mathcal Q$$-index of $$T$$ is defined by means of the dual lamination of $$T$$. It is a generalisation of the Poincaré Lefschetz index of a foliation on a surface. We prove that the $$\mathcal Q$$-index of $$T$$ is bounded above by $$2N-2$$, and we study the case of equality. The main tool is to develop the Rips machine in order to deal with systems of isometries on compact $$\mathbb R$$-trees. Combining our results on the $$\mathcal Q$$-index with results on the classical geometric index of a tree, developed by D. Gaboriau and G. Levitt [Ann. Sci. Éc. Norm. Supér. (4) 28, No. 5, 549-570 (1995; Zbl 0835.20038)], we obtain a beginning classification of trees.

##### MSC:
 20E08 Groups acting on trees 20E05 Free nonabelian groups 20F65 Geometric group theory
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##### References:
 [1] M. Bestvina and M. Feighn, Outer limits. Unpublished, 1994. · andromeda.rutgers.edu [2] M. Bestvina and M. Feighn, Stable actions of groups on real trees. Invent. Math. 121 (1995), 287-321. · Zbl 0837.20047 · doi:10.1007/BF01884300 · eudml:144300
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